1^,:;.  MATHEMATies  UBRAAl 

TEACHER'S  Manual 


First-Year  Mathematics 


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THE  UNIVERSITY  OF  CHICAGO 
MATHEMATICAL  SERIES 

ELIAKIM  HASTINGS  MOORE 
GENERAL  EDITOR 


SCHOOL  OF  EDUCATION 
TEXTS  AND  MANUALS 


GEORGE  WILLIAM  MYERS 
EDITOR 


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in  2017  with  funding  from 

University  of  Illinois  Urbana-Champaign  Alternates 


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TEACHER’S  MANUAL  FOR 
FIRST-YEAR  MATHEMATICS 


THE  UNIVERSITY  OF  CHICAGO  PRESS 
CHICAGO,  ILLINOIS 


THE  CAMBRIDGE  UNIVERSITY  PRESS 

LONDON  AND  EDINBUKGH 

THE  MARUZEN-KABUSHIKI-KAISHA 

TOKYO,  OSAKA,  KYOTO 

KARL  W.  HIERSEMANN 

LEIPZIG 

THE  BAKER  & TAYLOR  COMPANY 

NEW  YORK 


TEACHER’S  MANUAL 

FOR 

First- Year  Mathematics 


By 

GEORGE  WILLIAM  MYERS 

Professsor  of  the  Teaching  of  Mathematics  and  Astronomy,  College  of  Education 
of  the  University  of  Chicago 
and 

WILLIAM  R.  WICKES  ERNEST  L.  CALDWELL 

ERNST  R.  BRESLICH  ROBERT  M.  MATHEWS 

WILLIAM  D.  REEVE 

Instructors  in  Mathematics  in  the  University  High  School 
of  the  University  of  Chicago 


School  of  Education  Manuals 
Secondary  Texts 


THE  UNIVERSITY  OF  CHICAGO  PRESS 
CHICAGO,  ILLINOIS 


Copyright  1914  By 
The  University  of  Chicago 


All  Rights  Reserved 

Published  July  1911 
Second  Edition  August  1914 


Composed  and  Printed  By 
The  University  of  Chicago  Press 
Chicago,  Illinois,  U.S.A. 


S 10>‘l 

Cl 


PREFACE 

This  Manual  is  for  teachers  for  whom  the  phrase 
‘^modernized  mathematics  for  secondary  schools”  means 
something  more  than  a rallying  sentiment  for  teachers’ 
meetings.  It  is  for  teachers  who  see  in  this  phrase  a real 
meaning,  serious  enough  to  entitle  it  to  actual  recognition 
in  practical  teaching.  It  is  of  course  not  an  independent 
composition  for  continuous  reading;  but  is  for  the  use  of 
those  who  are  employing  First-Year  Mathematics  as  a text. 

The  text  on  which  this  Manual  comments,  presents  in 
teachable  form  an  interweaving  of  the  more  concrete  and 
the  easier  phases  of  the  first  courses  in  both  algebra  and 
geometry.  The  chief  emphasis  is  placed  on  algebra,  but  a 
considerable  body  of  related  fundamental  notions  and  prin- 
ciples of  geometry  is  woven  in,  while  some  advances  are 
made  in  one  direction  or  another  upon  the  most  concrete, 
graphic,  and  practical  aspects  of  elementary  geometry. 
Geometrical  treatments  are  at  first  intuitive,  inductive,  and 
experimental,  but  in  some  instances  these  are  followed  by 
the  quasi-experimental  methods  of  superposition.  A like 
informality  as  to  method  characterizes  the  earlier  part  of  the 
algebra.  The  transition  from  the  informal  procedure  of  the 
earlier  part  to  the  formal  procedure  of  the  later  part  is 
gradual.  The  reasoning  becomes  more  and  more  highly 
deductive  throughout  the  latter  half  of  the  book.  This 
accords  with  both  classroom  practice  and  a priori  reasoning 
as  to  the  normal  procedure  for  secondary  mathematical 
instruction  and  has  sound  historical  warrant. 

The  Manual  undertakes  to  do  two  specific  things,  viz. : 

First,  it  seeks  to  give  as  accurately  as  may  be  in  cold 
type  the  educational  and  mathematical  points  of  view  under 

vii 


685489 


viii  Preface 

which  the  authors  attempted  to  organize  the  material  of  the 
context;  and 

Second,  to  bring  to  the  reader  as  large  a measure  as 
possible  of  the  benefit  of  the  classroom  experience  of  the 
authors  who  have  been  using  First-Year  Mathematics  as  a 
text  with  high-school  classes. 

It  is  no  reflection  upon  the  professional  dignity  and  inde- 
pendence of  a teacher  to  feel  the  need  of  pertinent  pedagogi- 
cal suggestion  just  at  the  critical  point  where  the  situation 
is  opportune.  Callousness  in  this  regard  is  much  less  com- 
mon and  less  popular  now  than  it  was  a few  years  since. 
The  greatest  weakness  of  many  teachers  is  not  that  they 
know  no  pedagogy,  but  rather  that  what  they  have  is  poorly 
timed.  They  do  not  get  it  into  working  order  until  the 
critical  moment  has  passed.  The  night  after  the  unsatis- 
factory recitation,  or  the  next  day,  we  see  so  clearly  the  true 
pedagogic  key  to  the  entrance,  after  we  have  forced  passage 
by  clumsy  means.  Any  instrumentality  that  even  aids  us 
in  knowing  what  we  know,  just  when  we  need  it,  cannot 
fail  of  a cordial  welcome.  This  Manual  hopes  for  at  least 
friendly  recognition  on  some  such  score  as  this. 

The  text,  First-Year  Mathematics,  though  adhering  to 
algebra  as  a central  theme  of  first-year  work,  departs  per- 
haps far  enough  from  the  prevailing  type  of  text  for  beginning 
students  in  high-school  mathematics  to  justify  the  existence 
of  this  commentary.  The  authors  would  accordingly  have 
teachers  regard  this  little  volume  as  in  the  nature  of  a request 
for  the  privilege  of  a hearing  in  a case  in  which  perhaps  the 
burden  of  proof  is  upon  them,  rather  than  as  an  intimation 
of  any  widespread  need  of  teachers  of  secondary-school 
mathematics  for  general  instruction  in  mathematical  peda- 

gogy- 

With  one  exception  the  participants  in  the  authorship 
of  these  notes  and  suggestions  are  teachers  of  high-school 


Preface 


IX 


classes,  and  their  recommendations  come  warm  from  the 
classroom.  What  is  here  put  down  for  the  aid  of  others  may 
then  be  regarded  as  having  been  tested  and  seasoned  by 
trial,  and  not  at  all  as  a body  of  dogmatic  stipulations  either 
of  what  may  conceivably  be  done  with  high-school  classes,  or 
of  what  a narrow-gauge  pedagogy  would  prescribe.  Being 
merely  transcripts  with  emendations  under  expert  trial  and 
critical  scrutiny  of  what  the  authors  themselves  find  most 
practicable  in  administering  the  new  type  of  subject-matter, 
it  is  believed  that  these  notes  will  be  of  very  particular 
value  to  others.  The  prime  aim  in  their  preparation  has  of 
course  been  not  so  much  rhetorical  completeness,  as  real, 
practical  helpfulness  to  teachers. 

The  text,  Second-Y ear  Mathematics,  published  in  June, 
1910,  is  designed  to  continue  the  type  of  mixed  mathematics 
of  the  first  book  through  the  second  high-school  year,  though 
here  with  primary  emphasis  on  geometry.  The  two  books 
together  cover  considerably  more  than  the  essentials  of  what 
is  commonly  required  as  first-year  algebra  and  second-year 
plane  geometry.  This  leaves  the  pupil  in  condition  to  take 
up  any  of  the  current  texts  of  algebra  or  geometry  at  this 
point  of  the  curriculum. 

The  numbers  on  the  inside  top  corners  of  the  pages  of 
this  Manual  are  to  assist  the  reader  in  quickly  co-ordinating 
the  matter  given  here  with  the  relevant  material  of  the  text- 
book. On  the  left  page  of  each  pair  of  pages  lying  open  to 
the  reader  at  any  time  is  indicated  the  placing  of  the  earliest 
material,  and  on  the  right  the  latest  material  of  the  text 
that  is  here  being  discussed.  This  will  enable  users  of  the 
Manual  to  find  readily  in  both  text  and  Manual  the  subject- 
matter  under  consideration.  Of  course  the  value  of  what  is 
said  in  this  book  depends  entirely  upon  getting  it  readily 
co-ordinated  with  the  subject-matter  of  the  text  to  which  it 
appertains. 


The  Authors 


TABLE  OF  CONTENTS 

CHAPTER  PAGE 

Preface . vii 

Comments  on 

I.  General  Uses  of  the  Equation  . . . i 

II.  Uses  of  the  Equation  with  Perimeters  and 

Areas 8 

III.  The  Equation  Applied  TO  Angles  . . 17 

IV.  Positive  and  Negative  Numbers  ...  26 

V.  Beam  Problems  in  One  or  Two  Unknowns  37 

VI.  Problems  in  Proportion  and  Similarity  . 46 

VII.  Problems  on  Parallel  Lines:  Geometric 

Constructions  . 58 

VIII.  The  Fundamental  Operations  Applied  to 

Integral  Algebraic  Expressions  . . 63 

IX.  Practice  in  Algebraic  Language,  General 

Arithmetic 88 

X.  The  Simple  Equation  in  One  Unknown  . 99 

XI.  Linear  Equations  Containing  Two  or 
More  Unknown  Numbers;  Graphic  Solu- 
tion of  Equations  and  Problems  . .110 

XII.  Fractions 120 

XIII.  Factoring,  Quadratics,  Radicals  . . 127 

XIV.  Polygons,  Congruent  Triangles,  Radicals  151 


XI 


CHAPTER  I 


GENERAL  USES  OF  THE  EQUATION 

The  idea  of  the  equation  is  found  in  the  oldest  mathe- 
matical manuscripts.  As  a tool  for  problem-solving  the 
equation  is  not  only  time-honored,  but  it  is  also  of  such 
usefulness  and  importance  that  many  consider  that  the  main 
business  of  first-year  high-school  algebra  is  to  learn  what  the 
equation  is  and  how  to  solve  it  in  its  varied  forms.  Many 
problems,  very  difficult  by  arithmetic,  become  simple  and 
easy  by  means  of  the  equation.  It  is  long  enough,  if  not 
too  long,  to  wait  to  the  high  school  to  become  acquainted 
with  this  powerful  instrument.  Most  of  the  topics  of  first- 
year  algebra  can  be  seen  by  the  pupil  to  be  necessary  in  order 
to  use  the  equation  with  skill  in  solving  problems  of  one  sort 
or  another.  It  seems  natural  therefore  to  begin  a course 
in  mathematics  with  the  equation.  Another  reason  for 
introducing  the  equation  early  is  the  fact  that  it  is  a splendid 
instrument  to  arouse  interest  in  the  mathematics. 

Careful  teachers  know  well  from  experience  that  the 
concept  of  the  equation  is  not  so  easily  grasped  by  the  be- 
ginner as  is  commonly  assumed.  It  is  likely  to  be  conceived 
as  a mere  mechanical  form,  lacking  all  content,  and  requiring 
for  its  solution  merely  a sequence  of  mechanical  steps  leading 
in  some  mysterious  way  to  an  answer.’’  The  result,  if  it 
agrees  with  that  in  the  book,  of  these  mechanical  steps  that 
are  usually  imperfectly  memorized,  is  looked  upon  as  the 
solution  of  the  problem  without  further  question.  The 
only  question  in  the  learner’s  mind  is,  “Did  I remember  to 
take  the  steps  correctly?”  Such  empty  mechanical  imita- 
tion as  this  both  kills  interest  and  arrests  mathematical 


I 


2 


First-Year  Mathematics  Manual  [ pp,  i- 


development.  A clear,  even  though  not  a logically  com- 
plete, concept  of  the  signification  of  an  equation  forestalls 
much  of  this  thoughtless  manipulating  and  symbol- juggling. 
A good  start  is  half  the  battle.  Therefore  at  first  the  equa- 
tion is  regarded  as  stating  the  balance  between  two  number- 
expressions.  The  first  two  pages,  read  and  freely  discussed 
by  the  class,  the  teacher  acting  as  moderator,  will  give  clear 
ideas  about  the  equation  and  a satisfactory  working  hold 
upon  it. 

The  chapter  contains  a large  number  of  problems  and 
exercises.  It  will  be  found  unnecessary  to  work  all  of  them. 
Let  a number  of  problems  in  a certain  set  be  read  by  the  class, 
sufficient  to  give  the  class  clear  conceptions  and  a little 
practice  in  solving  them.  It  is  worse  than  useless  to  insist 
on  thoroughness  beyond  a very  low  limit.  Clearness  and 
sufficiency  are  the  desiderata  at  the  beginning.  During  or 
after  this  reading,  other  problems  in  the  set  may  be  assigned 
for  home  work.  If  there  are  still  problems  left  that  may  be 
omitted,  and  if  further  drill  is  felt  to  be  unnecessary,  the 
class  may  take  up  the  next  set  of  problems.  Thus  an  assign- 
ment for  home  work  may  contain  problems  taken  from 
several  pages. 

It  is  well,  while  teaching,  to  keep  in  mind  the  conclusions 
stated  at  the  end  of  the  chapter,  to  guarantee  arriving  at 
definite  results.  Verbal  problems  and  formal  problems 
should  be  mixed  freely,  that  the  study  of  neither  may  become 
monotonous.  No  attempt  should  be  made  to  master  every 
problem  or  to  work  with  over-mature  notions  or  thorough- 
ness. Sufficiently  clear  intuitional  knowledge  must  for  a 
time  take  precedence  of  thoroughness. 

The  chapter  is  to  serve  as  an  introduction  to  the  possi- 
bilities of  the  subject,  to  impress  the  pupil  with  its  usefulness, 
and  to  point  out  the  importance  of  the  equation.  The  main 
purpose  of  the  formal  equations  is  to  review  arithmetic  under 


-j]  General  Uses  of  the  Equation  3 

the  guise  of  algebra.  Pupils  profit  more  by  such  reviews 
than  by  frank  arithmetical  work  of  which  they  are  tired. 
Nine  recitations  of  45  minutes  each  is  the  maximum  amount 
of  time  to  be  spent  on  it. 

Lesson  i:  pages  i-j,  Problems  Using  Algebraic  Language 

Let  a pupil  read  aloud  carefully  Problem  i,  and  answer 
the  question.  Then  read  the  algebraic  solution  and  make 
clear  that  the  value  of  w is  found  by  subtracting  8 from  both 
sides  of  the  equation. 

Problem  2 is  read  next  by  another  pupil.  This  brings 
out  the  same  point  as  Problem  i,  but  in  addition  calls  for  a 
solution  of  2p  = i6,  The  value  of  p is  found  by  reasoning 
thus:  If  twice  p is  16,  then  is  | of  16,  or  8. 

Problem  3 brings  out  the  same  point  as  do  Problems  i 
and  2. 

Problems  4,  5,  6 show  how  the  value  of  x is  found  by 
adding  the  same  number  to  both  sides  of  the  equation. 

All  these  problems  are  to  aid  the  pupil  to  convert  his 
arithmetical  way  of  thinking  into  its  algebraic  form.  The 
exercises  below  have  the  same  office. 

A pupil  reads  § 2,  page  3.  This  is  followed  by  a short 
discussion  both  by  teacher  and  class. 

Solve  by  inspection  problems  of  Exercise  I,  occasionally 
putting  a solution  on  the  board  as  the  pupil  is  reading  it. 
Assign  for  Lesson  2 all  of  pages  i and  2 and  some  problems 
in  Exercise  I. 

Solutions:  Exercise  I 

No.  ofprob.:  i,  2,  3,  4,  5,  6,  7,  8,  9,  10,  ii, 

Solution:  2,  5,  5,  4,  5,  7,  16,  7,  4i,  6,  3, 

No.  ofprob.:  12,  13,  14,  15,  16,  17,  18,  19,  20,  21 

Solution:  2^,  9,  6,  8,  7§,  9,  6,  12,  8,  9 


4 


First-Year  Mathematics  Manual 


[PP-  3- 


Lesson  2 : pages  3-4,  Problem  36,  Exercise  II,  included 

Take  up  5 to  10  minutes  answering  questions  asked  by 
the  class  and  asking  questions;  and  adding  explanation,  if 
needed,  to  the  work  covered  in  Lesson  i. 

Let  the  class  read  and  discuss  Problems  i and  2,  page  3. 
Assign  3 on  page  4 to  be  worked  in  a similar  way.  Let  the 
class  work  orally  most  of  the  problems  in  Exercise  II. 
Assign  for  home  work  Problems  34,  35,  36.  Problems  of 
Exercise  II  give  drill  work  in  mental  arithmetic,  and 
assist  the  pupil  in  getting  hold  of  the  idea  of  literal  number. 
They  also  give  further  practice  in  changing  from  the  arith- 
metical to  the  algebraic  way  of  thinking. 


Solutions:  Exercise  II 


No.  of  prob.: 

: I, 

2, 

3? 

4, 

s. 

6, 

7, 

. 8, 

9, 

Solution : 

27, 

56, 

42, 

72, 

63, 

63, 

42 

, 54, 

96, 

No. 

10, 

II, 

12, 

13, 

14, 

IS, 

16, 

17, 

18, 

19, 

Sol. 

96, 

72, 

6, 

7, 

4, 

8, 

97 

8, 

9, 

9, 

No. 

20, 

21, 

22, 

23, 

24, 

25, 

26, 

27, 

28, 

29, 

Sol. 

7. 

7, 

6, 

6, 

12, 

63, 

35, 

72, 

II, 

II, 

No. 

31, 

32, 

33, 

34, 

3S> 

. 36, 

37, 

38 

Sol. 

1 

7) 

1 

87 

1 

6> 

1 

iy 

I . 

2, 

27,  I 

5+1^ 

Lesson  3 : pages  4-10,  Problems  and  Exercises 

Begin  as  in  Lesson  2.  Let  the  class  read  from  Problems 
37,  38,  page  4,  to  Problem  4,  page  5.  These  problems  bring 
out  the  idea  of  general  number  still  more  sharply  than  did 
the  preceding  problems.  Let  the  class  read  carefully 
Problems  i,  2,  3,  at  the  bottom  of  page  5.  Then  follow  with 
reading  and  suggestions  as  to  the  method  of  working  Prob- 
lems 3,  4,  on  page  6.  Assign  3,  4,  5 for  the  next  day.  Use 


-(5] 


General  Uses  of  the  Equation 


5 


the  remaining  time  to  solve  orally  about  15  problems  in 
Exercise  III,  p.  9. 

Lesson  4:  pages  q-io 

After  opportunity  is  given  for  asking  questions  and  a 
short  review,  let  the  class  read  Problems  6-9,  pages  6,  7,  and 
the  meaning  of  all  problems  should  be  made  clear  and  sug- 
gestions be  given  as  to  the  ways  of  obtaining  the  equations. 
Have  the  pupils  solve  orally  or  at  the  board  Problems  16-30, 
page  9,  and  assign  for  home  work  every  third  problem  of 
Nos.  31-50  on  page  10. 

Lesson  5:  pages  g-ii 

This  will  be  much  like  the  two  preceding  lessons,  cover- 
ing Problems  9,  10,  ii,  12,  page  7,  Problems  i,  2,  3,  page  10, 
and  Problems  6-36,  page  ii. 

Lesson  6:  Problems  ij  to  17,  page  7;  Problems  4 and  5, 
page  10;  Problems  37  to  55,  page  ii 

Lesson  7 : Problems  18  to  23,  page  8;  Problems  i to  5, 
page  12 

Lesson  8:  Problems  24  to  30,  page  8;  Problems  6 to  13, 
page  12 

Review  and  conclusions,  page  13. 

Lesson  9:  Written  Exercise  on  Chap,  i 

Solutions  of  Problems:  from  page  5 to  page  13 
Page  5 Page  6 

No.  ofprob.:  i,  2,  3,  4 3,  4,  5,  6,  7 

Solution:  14,  22-a,  a-b,  3a  15,  5,  40,  8,  2 

30,  20,  20,  14,  10 

25,  80,  10,  108 

24,  40, 

IS, 


6 


First-Year  Mathematics  Manual 


[ pp.  7- 


Page  7 


No.: 

8, 

9,  10,  II, 

12, 

13. 

14, 

15 

Sol.: 

No.: 

32, 

24, 

20, 

24, 

32, 

20, 

i6. 

36.  21  105,  S2j,  3 hr., 

12,  rd.,  210,  262I,  150  mi. 

9) 

Pages  8 and  9 

17,  18,  19,  20,  21, 

24, 

) SI) 

22, 

503 

528 

553 

23 

Sol.: 

4, 

53i  9.  7. 

1.641. 

95. 

lOI, 

157 

4, 

no.  33. 

3.606, 

121, 

102, 

158 

12, 

159 

12, 

52, 

No.: 

24, 

25.  26, 

27. 

28, 

29, 

30 

Sol.: 

77. 

188,  122, 

40, 

10, 

42, 

100 

79. 

190,  124, 

180, 

210, 

60 

126, 

400, 

89. 

Solutions:  Exercise  III 

Page  9 

No.  of  prob.: 

I,  2,  3, 

4.  5. 

6, 

7.  8, 

9) 

Solution: 

3.  4.  6, 

10,  8, 

9) 

8,  7. 

3) 

No.: 

10,  II,  12,  13,  14 

. 15. 

16,  17,  18, 

19) 

Sol.: 

6,  12,  9,  8,  84,  9, 

9,  7.  9. 

7. 

No.: 

20,  21 

, 22,  23,  24, 

25.  26, 

27, 

28,  29, 

30 

Sol.: 

12,  2§ 

. 7.  10,  12, 

4,  12, 

9) 

2,  6, 

7 

Page  10 

No.: 

31.  32, 

33.  34.  35.  36, 

. 37.  38,  39. 

40,  41. 

42, 

Sol.: 

10,  4, 

4i.  7.  3.  6, 

7,  2 

. 7. 

3i.  3. 

5. 

No.: 

43)  44, 

45.  46,  47.  48, 

49.  50, 

I,  2, 

3.  4, 

5 

Sol.: 

24.  6, 

12,  24,  4,  14, 

16, 24, 

4,  3) 

2,  -I, 

— 2 

12,  II, 

5.  4. 

4 

6o, 


-12]  General  U ses  of  the  Equation  7 


Page  II 


No.  of  prob.: 

6, 

7, 

8, 

9, 

10, 

II, 

12, 

13, 

Solution : 

2^  43, 

3i, 

2f, 

7i 

6i, 

9, 

III, 

No.: 

14, 

15, 

16, 

17, 

18, 

19, 

20, 

21, 

22, 

23, 

Sol.: 

7i 

6§, 

sl. 

4^, 

7i 

04, 

04, 

sl. 

41, 

sl. 

No.: 

24, 

25, 

26, 

27, 

28, 

29, 

30, 

31 

, 32, 

33, 

Sol.: 

St, 

n— 

97, 

s§. 

3i 

8, 

45, 

10, 

6, 

4, 

8.S, 

No.: 

34, 

35, 

36, 

37, 

38, 

39, 

40, 

41, 

42, 

43, 

Sol.: 

II, 

10, 

10, 

8, 

12, 

II, 

4, 

II, 

12, 

4, 

No.: 

44, 

45, 

46,  47 

, 48. 

. 49, 

50, 

51, 

52, 

S3,  54, 

'_55, 

Sol.: 

tV, 

5, 

4,  60 

,.  IS, 

• 56, 

42, 

48, 

3, 

1 1 
3,  4, 

i|. 

Page  12 


No.  of  prob.:  i, 

2, 

3, 

Solution : xf  1 20 

t^|oXi2o 
'^iV|Xi2o 
TwX  120 

r|-o-X25 

rwX25o 

rfoXi527 

rlrXi 

rSu'X2o 

rJ^XSo 

rSirXft 

No.: 

6, 

7, 

8, 

Sol.: 

175XTI7X2 

i75XtItX5 

i75XToirXf 

175X1-^X21 

i75Xto7X< 

600X5  XxtiF 
booXsXxtir 
6ooXsXtw 
booXsX^W 
booXsXxTjy 

i6oX^X  100 

i6oX^X 

i6oX^Xtw 

No.: 

9, 

10, 

II 

Sol.: 

rl:TX3X2oo 

1^X3X360 

t^X3X7S6 

oX3X^ 

rl^X^X^ 

rwX/X/>. 

{pp.  14- 


CHAPTER  II 

USES  OF  THE  EQUATION  WITH  PERIMETERS  AND 
AREAS 

This  chapter  is  to  be  covered  in  not  more  than  6 lessons. 
The  summary  at  the  end  states  the  most  important  results 
to  be  obtained. 


Lesson  i : pages  14,  75, 16,  and  17 


Problems  i to  7 should  be  read  by  the  class.  They  should 
help  to  make  the  pupil  see  clearly  that  a+b  is  both  a number 
and  the  sum  of  a and  b.  Pupils  usually  feel  that  only  terms 
having  a common  factor  can  be  added  and  do  not  consider 
a-\-b  as  a sum.  Let  the  class  read  Problems  8 to  15,  that 
the  pupil  may  feel  that  numbers  having  a common  factor 
have  a sum  expressed  as  one  term. 


Solutions  of  Problems:  pages  14  and  15 


1.  8+5 

2.  a+^ 

7.  (i)  x-^ 


(2)  a— 12 

(3)  y-io 

(4)  x-y 


3.  m+p 

4.  18-7 

(5) 

(6)  15-M 

(7)  r-s 

(8)  a—x 


6.  m—n 

( 9)  x—a 

(10)  d—c 

(11)  c—b 

(12)  t—m 


5.  w-8 


8.  (12+10)  yd. 

9.  (8+4)  doz. 

10.  (s+4)  12’s 

11.  (9+4)Xi2 


12.  (8+6)  half-dozens 


13.  (8+6)X6 
14*  (S+7+io)X3 
IS-  (2+3)^7  2X,  4X. 


8 


Equation  with  Perimeters  and  Areas 


9 


-i6] 


§§  5>  6,  and  7 give  drill  on  the  meaning  of  the  terms 
“exponent’’  and  “coefficient,”  and  make  clear  the  distinc- 
tions between  them.  Use  more  illustrations  if  needed. 
These  terms  should  be  used  freely  after  this. 

§ 8 brings  out  this  difference  still  more  sharply.  By 
substituting  values  the  pupil  is  made  to  see  that  is  a 
product  containing  3 factors  x.  All  12  problems  should  be 
taken;  or  some  may  be  selected  and  various  values  for  x 
substituted.  Some  may  be  assigned  for  home  work.  Call 
attention  to  the  difference  between  the  meaning  of  and 
(3a;)". 

Solutions 


I. 

12 

7- 

30 

2. 

36 

8. 

6 • 6 ■ 

• 6 • 6 • 6 

3- 

18 

9- 

72 

4- 

6 • 6 ■ 

• 6 

10. 

108 

5- 

24 

II. 

5 . 6 ' 

•6*6 

6. 

6 • 6 ■ 

•6*6 

12. 

2 • 6 • 

• 6 • 6 • 6 

Lesson  2 : pages  16  to  middle  of  18 

This  lesson  brings  in  some  of  the  terms  used  in  beginning 
geometry.  It  shows  how  geometric  concepts  can  be  repre- 
sented algebraically,  and  conversely,  that  an  algebraic 
expression  may  be  thought  of  as  symbolizing  the  measures 
of  geometric  magnitude.  Thus  we  have  algebraic  numbers 
representing  line-segments,  e.g.,  sides  of  triangles,  perim- 
eters (see  Problems  i to  7)  algebraic  products  representing 
areas  of  squares  and  rectangles.  These  are  important  as 
adding  to  the  pupil’s  conviction  that  these  letters  are  num- 
bers. Only  a little  time  need  be  taken  to  read  and  answer 
Problems  i to  8.  The  value  of  the  exercises  will  be  much 
greater  if  the  work  moves  right  along  briskly  than  if  it  is 
permitted  to  drag.  Problem  i,  page  17,  should  be  taken  up 
as  a review  and  extension  of  what  has  been  learned  on  page 


lo  First-Year  Mathematics  Manual  [pp.  i6- 

15.  They  aid  the  pupil  to  understand  that  a product  has 
a concrete,  picturable  meaning.  All  12  problems  should  be 
thought  of  as  products  in  which  the  factors  are  counted. 
Thus,  4*  • 4^  = 4^  since  there  are  4 factors  4 in  the  product. 
Problems  2 to  9 call  for  a statement  for  finding  the  area  of 
the  square  and  of  the  rectangles  and  give  drill  in  using  the 
equation  and  stating  algebraic  products. 


Soluhons:  page  16 


I.  ^x^t.  2.  13a 

§9.  I.  I perimeter,  (2X+2y) 

2.  p = s+S+S+S 
A=S  • 8 

3.  p = 4 • 12,  ^ = I2» 
p = 4.  • s,  A =s^ 

p = 4.  • X,  A =x^ 
p = 4 • a,  A=a^ 

P = A • y,A=y^ 

4.  side  = 4,  ^ = 16 
side  = a,  p = ^a 


5.  side=5,  ^ = 25 
side  = x,  A=x^ 

7.  p = 2(a+l?),  ^p  = a+by 


8.  A=a  • b. 

§10.  I.  (i)  4'' 

(s) 

(2)  8^ 

(6) 

(3)  10^ 

(7) 

(4)  127 

(8) 

2.  p-^X2,x 

^ = 3^*3^ 


3.  3xrd, 


p = Sb,A=4b^ 

/?  = 4(io+2),  A =(10+2)^ 
p = 4{b+7),  A = (b+7y 

p = 4{cA-d)j  A = {c-]rdy 

side  = 2a,  p = Sa 
side  = x,  p = 4x 

side  = 2a,  A =4a^ 

2a,  2b 


a^ 

(9) 

b^ 

a^ 

(10) 

a^ 

(ii) 

a” 

(12) 

Equation  with  Perimeters  and  Areas 


II 


-i8] 


Find  area  by  counting  squares. 

3.  ^=8  . s,  8 . 4,  8 • 3i,  8 . 6i 

4.  ^ = 12-6,  12  • 8i,  12  • 9^,  12  • lof,  12X,  i2y 


S* 

6. 

7* 


A = i2  • 9 • /,  h • I,  n • I,  X • I,  a • I 

A=SWj  loWy  i2^w,  wx,  aWj  Iw,  bw,  wz 
a • X =ax  b • b^  = b^  x^  • x =x^ 


b • c =bc 
b -b^b^ 


a^  • a =a^ 
a^  • a^  = a^ 


a • x^  = ax^ 

a^  • X =a^x 


a • a^  = a^ 


X • x^  = x^ 


g • 


8.  Count  squares. 

9,  Draw  figure  and  count  squares. 


10.  Transform  parallelogram  into  an  equal  rectangle. 
Then  ^ = ioa,  i2\a,  16. 8a,  ab,  ac,  ax,  2an,  ^az. 


II.  Show  that  the  area  of  triangle  is  one-half  the  area 
of  the  parallelogram.  A = ^ • 12a,  J • i8^a,  | • 20.25a, 
\ab,  \ad,  \ax,  \a  • 2y,  | • 42a,  \ • ^ac. 


Lesson  3:  pages  18,  20,  one-half  of  21 

Recall  how  the  area  of  a rectangle  is  obtained.  Have 
the  pupil  see  clearly  how  the  rectangle  representing  6x^  is 
made  up  of  six  squares,  each  x^.  Do  not  slight  this  visualiz- 
ing. Point  out  that  there  are  formulae  for  finding  the  areas 
of  other  geometric  forms  as  Figs.  12,  13,  14,  and  16.  Show 
how  the  area  of  the  parallelogram  is  found  by  means  of  a 
rectangle  equal  to  it  in  the  obvious  fashion  suggested  in 
Fig.  12.  Solve  Problem  10.  Knowing  how  to  find  the 
area  of  the  parallelogram  we  are  now  able  to  find  the  area 
of  the  triangle  (Fig.  13).  Solve  Problem  ii.  Do  not  let 
the  work  drag.  Action  and  movement  are  needed  here  else 


12 


First-Year  Mathematics  Manual  [pp.  i8- 


the  metal  cools.  In  a similar  way  work  Problems  12  to  15. 
Let  the  class  read  and  answer  Problem  16.  Do  not  multiply 
out.  Simply  have  the  equation  stated  thus:  A=6{x-\-s)y 
^(^+5)j  a{x—2),  (aH-2)(x+i);  Problem  17:  A = {m+n) 
But  mcA-md-\-nc-{-nd,  Problem  18,  represents  the 
same  area.  Therefore,  Problem  19,  {m-{-n){c-\-d)=mc-{- 
md+ncA-nd.  You  can  readily  convince  yourself  here  that 
the  learner  does  not  naturally  reason  with  axioms  as  he  will 
not  find  19  so  easy.  But  deductive  reasoning  must  be  ac- 
quired and  for  this  the  profit  is  in  the  work.  Solve  Prob- 
lems 20  (i);  21  (i);  22  (i)  and  (2)  in  class,  making  sure  the 
work  is  understood  by  the  class,  and  assign  for  home  work 
the  remaining  parts  of  Problems  20  to  22. 


Solution  for  Problem  20 

(1)  xaA-yd-Y^bA-yb  (3)  abAroob+dyA-^y 

(2)  (4)  raA-saA-rxA-sx 

The  figures  for  Problem  21  are  all  squares  of  sides  a+6, 
c+c?,  xA-y,  mA-n  respectively. 


Solution  for  Problems  22  and  25 

22.  (i)  {a-\-xy  (3)  {k-yby  (s)  (x+3)^ 

(2)  (6+c)"  (4)  (^+0"  (6)  {c+^y 

40 

23-  « = ^ = 5 

Let  the  class  read  rapidly  Problems  24  to  26,  assigning 
the  last  of  Problem  26  for  home  work.  Do  not  stop  to  multi- 
ply the  quotients;  simply  state  the  equation,  e.g.,  a — ^^~ 
(read  32  over  8).  The  problems  on  page  21  are  to  be  read 
in  a similar  way. 


13 


-24\  Equation  with  Perimeters  and  Areas 

Solutions  for  pages  20  and  21 


24.  a = — , — , — , — . Take  time  to  simplify  only 
0065 

some  of  the  fractions. 


25.  b = 

9 ' 9 ' 9 
some  of  the  fractions. 


27  18  15  12  ^ ^ ^ . ...  , 

— 1 ake  time  to  simplify  only 


24  12  Q 

26.  a = i — 7 , 1 — 7 , 1 / , i“A  • Take  time  to  simplify 

2*0  2*h  2*h  2*^ 

only  some  of  the  fractions. 

16  8 4 a hi  ah  by  a h 

7 ’ I 7 ’ i 7 T7’  Ia’  Ia’  17’  17* 

2*4  2*4  2*4  2^  2^  2^  2^  2^  2^ 

Take  time  to  simplify  only  some  of  the  fractions. 


« _m  a b \x  a-\-b  a a+6  a^—b^  a^—b^ 

^ n ^b  ^ a^  ^ ^ cA-d^  a-\-b  ^ a—b  ^ 

x^  {a-\rby  {a— by 
a-\~b  ^ a-\~b  ’ a — b 


Lesson  4:  pages  21^  22,  2j,  and  one-half  of  page  24 

§§  12  to  18  may  be  read  by  the  class  or  may  be  pre- 
sented by  the  teacher,  giving  the  text  for  reference  at  the 
close  of  the  lesson. 

Solve  Problems  i to  12  with  the  class,  the  class  helping, 
keeping  the  work  moving  right  along  briskly.  Let  pupils 
think  about  Problem  14,  and  let  some  pupil  volunteer  to 
put  the  figure  on  the  board.  Similarly  for  Problems  15  to 
18.  Assign  19  to  22  for  home  work.  Have  the  class  solve 
part  of  Problem  23  and  assign  part  for  home  work.  Similarly 
for  Problems  25  to  37.  Discuss  in  class  problems  5 and  7 
of  § 19  and  assign  2 or  3 of  6 to  16  for  home  work. 


14 


First-Year  Mathematics  Manual  [pp,  22- 


Solutions 

§18.  I.  a • a • a. 

2.  4 • a • a • &. 

3.  Subtract  3 from  10  and  multiply  the  result  by  6. 

4.  8 times  the  sum  of  40  and  2. 

5.  9 times  the  difference  between  60  and  3. 

6.  7 times  the  sum  of  80  and  i. 

7.  16  times  the  sum  of  a and  b, 

8.  Subtract  y from  x,  add  i,  and  multiply  the  result  by  20. 

9.  25  times  the  difference  between  x and  a. 


Page  23.  I.  Equilateral  triangle,  equilateral  4-side,  etc. 

12.  Equilateral  aside, 

13-  3,  4,  5,  6,  7,  8,  9,  10,  12,  20,  n,  a. 

14.  A rectangle  of  dimensions  2X  and  *2. 

15-19,  21.  May  all  be  rectangles. 

20.  A triangle  all  sides  of  which  are  x+y^ 

22.  A triangle  whose  sides  are  2x+y,  x+y,  2x+y. 

23.  p=^6,  8,  10,  12,  14,  16,  18,  20,  24,  40,  2n,  2a] 

p = g,  12,  IS,  18,  21,  24,  27,  30,  36,  60,  sn,  3a; 
p = iS,  24,  30,  36,  42,  48,  54,  60,  72,  120,  6n,  6a, 

24.  p = 24,  34,  34,  36,  18,  36,  21,  24,  31,  52,  76,  76,  64, 

60,  120,  48,  56,  72;  4a+4,  6a+4,  6^1+4, 
4a+i6,  6a--i2,  12^  — 24,  3^+36,  4^+2^, 
5a+3^- 

25-32.  Figures  are  all  rectangles,  the  factors  being  the 
length  of  the  sides  (see  Fig.  21). 

3 

33-36.  Rectangles  of  dimensions  - , a:+i;  2,  a+i; 

b+c 

a,  ——  ; *+4  • 

2 

37.  9,  36,  8,  30,  18,  28,  o,  20,  6,  10,  4,  -3;  25,  90,  40,  40, 
54,  5,  36,  9,  16,  18,  5. 


-27l  Equation  with  Perimeters  and  Areas  15 

§ 19.  I.  Multiply  the  sum  of  x and  y by  the  sum  of  a 
and  10. 

2.  Multiply  the  sum  of  a and  b by  the  sum  of  c and  d, 

3.  Multiply  the  sum  of  x and  y by  the  difference  of  a 
and  b. 

4.  Multiply  the  difference  of  a and  b by  the  difference  of 
c and  d. 

Lesson  5 : page  24,  Section  20,  to  middle  of  page  27 

Ask  for  the  meaning  of  each  of  Problems  i to  14  before 
solving  or  assigning  for  home  work.  Solve  with  the  class 
Problems  i,  2,  4,  5,  8,  9,  12.  Assign  the  others  for  home 
work.  Let  the  class  read  page  25  to  the  middle  of  page  26. 


►.  I. 

12 

6. 

360 

II. 

1/24 

2. 

104 

7- 

8 

12. 

2 

3- 

12 

8. 

4 

13* 

1,000 

4- 

64 

9- 

3 

14. 

25,000. 

5- 

36 

10. 

3 

Lesson  5 : from  Section  20,  page  24,  to  middle  of  page  26 

Let  the  class  read  Article  21.  Have  problems  read,  let 
some  be  solved  in  class  and  the  others  assigned  for  home  work. 
Send  class  to  the  board,  assign  to  each  pupil  one  of  the  fif- 
teen problems  in  Exercise  IV,  have  some  of  these  problems 
explained  and  assign  others  for  home  work. 

Discuss  with  class  ways  of  solving  each  problem  before 
solving  or  assigning  for  home  work,  getting  as  much  sugges- 
tion as  possible  from  the  class.  Let  one  pupil  work  at  the 
board  with  the  class  criticizing  or  suggesting  better  ways. 
Always  and  everywhere  the  idea  is  to  work  with  the  class  not 
for  the  class.  Good  teaching  must  engage  the  learner  in  the 
work. 


First-Year  Mathematics  Manual  [pp,  27- 


Lesson  6 

From  § I,  p.  26,  to  the  Summary.  Review  the  problems  on  p.  28. 
Solutions:  page  26 


I. 

9 

(:x:+8)8  = 96 

2. 

12 

6. 

x+8=  12 

3- 

31 

x = 4 

4. 

25-5 

7- 

f 7 3 ? 5 > 3 > I ) 

Solutions:  Exercise  IV 


I. 

17 

7- 

5 

13- 

8 

19. 

7i 

25. 

10 

2. 

12 

8. 

6 

14. 

9 

20. 

4 

26. 

4 

3- 

6 

9- 

6 

IS- 

IS 

21. 

6 

27. 

4- 

7 

10. 

12 

16. 

4 

22. 

10 

28. 

_2_0. 

7 

5- 

7 

II. 

6 

17- 

IS 

23- 

S6 

29. 

¥- 

6. 

6 

12. 

18. 

32 

24. 

21 

30. 

S6 

-32] 


CHAPTER  III 

THE  EQUATION  APPLIED  TO  ANGLES 

Lesson  i : pages  30-34 

The  first  five  pages  teach  the  conception  of  angle  as  the 
amount  of  turning  and  give  practice  in  reading  and  drawing 
angles.  In  §24  and  §25  “angle’’  is  defined  as  the  amount 
of  turning  of  a line  rotating  about  a point  as  a pivot.  As  a 
familiar  illustration,  the  angle  formed  by  the  hands  of  a 
clock  may  be  used.  In  § 25  the  right  angle  is  defined  as  \ 
of  a complete  turn  and  the  straight  angle  as  | of  a turn. 
This  should  help  the  pupil  to  avoid  the  common  mistake  of 
calling  a straight  angle  a straight  line.  It  is  highly  important 
that  the  pupil’s  general  stock  of  everyday  notions  be  drawn 
upon  as  largely  as  possible  in  defining  the  basic  notions  and 
concepts  of  mathematics. 

Drawings  in  Problem  i page  31,  shall  be  like  those  in 
Figs.  31  and  32.  After  the  first  two  or  three  they  may  be 
merely  sketched  rapidly.  Some  of  these  may  be  put  on  the 
board  by  pupils,  others  assigned  as  home  work.  Problem  2 
should  be  read  and  answered  rapidly,  the  answers  given  to 
each  question  thus:  4 right  angles,  ^ of  4 R.A.,  7X4  R.A., 
1^X4  R.A.,  /X4  R.A.,  1X4  R.A.,  etc.  It  is  not  necessary 
to  simplify  answers  to  the  questions.  Problems  3,  4,  and  5 
may  be  read  in  a similar  way. 

Answers  for  Problem  3:  2 straight  angles,  1JX2  S.A., 
fX2  S.A.,  etc 3(2/  — 5)X2  S.A. 

Answers  for  Problem  4:  2 R.A.,  5X2  R.A.,  7^X2  R.A., 
etc (3  5 — 2)2  R.A. 

Answers  for  Problem  5:  qXi  S.A.,  2Xi  S.A.,  . . . . 

(4r-6)XiS.A. 


17 


i8 


First-Year  Mathematics  Manual  [pp,  j2- 


Problem  6 may  be  read  by  a pupil,  the  teacher  folding 
the  paper  as  indicated.  The  angles  are  right  angles  as  they 
are  equal  and  just  fill  the  plane  around  point  O. 

Let  different  pupils  read  each  a question  in  Problem  i, 
page  33,  and  give  the  answer,  as  ZXOA  = 3o°,  etc. 
Similarly  for  Problem  2.  The  answers  will  be  ZXOA+ 
ZA  OB  = 6s®,  etc.  In  Problem  3 the  answers  need  not  be 
simplified.  They  are  (2X180)°,  (4X90)°,  (fXgo)®,  (1X90)°, 


Problems  on  page  34  may  be 


assigned  for  home  work. 


Answers  to  questions:  i,  page  34:  2 R A.;  i S.A.;  180.° 
2,  page  34;  10°,  i R.A.;  30°,  | R.A.;  90°,  i R.A.; 
120°,  I R.A.;  150°,  f R.A.;  20°,  | R.A.;  80°,  f R.A.;  90°, 

1 R.A.;  120°,  I R.A.;  60°,  | R.A.;  170°,  V'  R-A.;  180°, 

2 R.A. 


3,  page  34:  ZXOB;  ZAOC;  ZXOY. 

4,  page  34:  ZX  O B=  ZX  O A+ZA  0 B . 

5,  page  34:  ZA  O C=  ZA  O B+ZB  O C , ZXOC  = 
ZXOY+ZYOC. 

6,  page  35:  ZXOC=ZXO  A+Z  AOB+ZBOC, 
ZAO C= ZAO B+ZB OY+ZYOC. 

7,  page  35:  Z X 0 B— ZX  O A=  ZA  O B. 

8,  page  35:  ZXOY-ZXOA=  ZAOY;  ZAOY- 
ZAOB=ZBOY;  ZX  O Y- ZB  0 Y=  ZX  0 B. 


Lesson  2 : pages  35-38 

Let  different  pupils  read  Problems  6,  7,  and  8 and  let  the 
class  give  the  answers  orally  to  all  questions  as  they  are 
read.  Work  out  with  the  class  the  first  two  angles  in  Prob- 
lem 9 and  let  the  table  be  filled  out  completely  at  home. 
Point  out  that  Problem  10  furnishes  a check  for  the  accuracy 


The  Equation  Applied  to  Angles 


19 


of  the  work.  Assign  for  home  work  Problems  ii  and  12. 
10,  page  35:  180° 

12,  page  36:  360°. 


Let  the  class  read  Problem  i,  page  36,  and  let  the  teacher  ask 
the  question  of  Problem  3.  Show  that  Problem  2 suggests 
a check.  In  all  this  work  some  things  are  to  be  done  by  the 
teacher  and  some  by  the  class.  The  danger  to  be  con- 
tinually guarded  against  is  that  the  teacher  will  do  too  much, 
thereby  carrying  his  class.  The  teacher  may  present 
Problems  4,  5,  6,  and  7 to  save  time,  questions  being  dropped 
to  the  class  now  and  then  to  hold  attention.  Let  the  class 
state  the  equation  for  some  of  the  problems  at  the  bottom 
of  page  37,  and  assign  some  of  them  for  home  work,  as  also 
those  on  38  and  in  Exercise  V as  a review. 

Page  37: 


§28.  I.  x=3i,93,  31,  56,  180 

2.  x=35 

70,  35,  180,  75 

3.  rr=37 

37,  221,  102 

4.  X = 24 

72,  120,  165,  3 


S-  ^=15 

24ii,  7i  III 

6.  x=io 

30,  76f,  20,  233I 

7.  x=9 

37,  i79f>  93f- 


Lesson  3 : pages  38-42 

Solve  as  many  of  the  problems  on  page  38  as  possible. 
The  arithmetic  of  the  number  combinations  is  the  important 
thing  here. 

Page  38:  8.  x = i4 

14,  70,  96 

9.  x=i3 

117,  13,  II,  39 

10.  X=I2 

96,  12,  38,  34 


20  First-Year  Mathematics  Manual  [pp.  j8- 

11.  x = S 

6sf,  726-,  48,  -6^ 

12.  X=24 

72,  66,  24,  18 

13-  2;=35 

70,  90,  17,  3 

14.  x=s7 
102,  70,  8 

IS-  2;=33 . 12 

132 . 066,  7 . 654,  40 . 28 

16.  x=g 

62 . 37,  43  • 38,  16 . 52,  57  . 73. 


Exercise  V,  page  38 


I. 

5 

7- 

2H 

13- 

9T7 

2. 

5 

8. 

6 

14. 

6 

3- 

12 

9- 

4 

IS- 

7h 

4- 

10 

10.  7f 

16. 

10 

5- 

S 

II.  16 

17- 

12 

6. 

2 

12.  15 

18. 

I. 

Assign  for  home  work  §§  29-31,  giving  some  suggestions 
as  to  the  nature  of  the  work;  but  not  telling  the  whole 
matter  and  leaving  no  real  work  for  the  class.  Your  pupils 
will  do  more  work  if  you  expect  it  of  them. 

Let  the  teacher  present  the  construction  of  Figs.  49-51. 
Let  all  members  of  the  class  make  the  construction  on  paper 
at  their  seats  under  the  suggestive  direction  of  the  teacher. 
Assign  for  home  work  portions  of  Problems  i to  4,  page  41. 
Make  clear  the  meaning  of  the  definitions  in  §§  33,  34. 
Paper  folding  and  creasing  are  good  here.  Let  some  pupil 
read  Problems  1,2,  and  3,  one  at  a time,  then  have  someone 
make  the  drawing  on  the  blackboard.  Explain  the  mean- 
ing, not  the  solution,  of  Problems  4 to  6,  page  42,  and  assign 


-^5] 


The  Equation  Applied  to  Angles 


21 


them  for  home  work;  one  at  a time  during  the  next  three 
days. 


Lesson  4:  pages  42-46 


Explain  the  meaning  of  the  definitions  in  § 35.  Paper 
folding  and  creasing  are  good  here  again.  Let  the  class 
work  Problems  i and  2.  Discuss  the  meaning  of  Problem  3 
and  assign  it  for  home  work.  Let  the  pupils  pin  on  ordinary 
notebook  paper  the  tracing  paper  with  the  drawing  and 
hand  this  in  with  the  other  home  work  assigned.  Let  the 
teacher  grade  it  and  hand  it  back.  The  answers  to  4 should 
come  rapidly  thus:  yes,  no,  yes,  yes. 

Let  the  class  work  orally  and  rapidly  all  problems  on 
page  43,  making  clear  the  point  that  the  equation  a+6  = 
180  is  the  algebraic  way  of  stating  that  angles  a and  b are 
supplementary.  Let  the  class  work  as  many  problems  of 
pages  44  to  46  as  possible,  and  assign  for  home  work  some  of 
the  verbal  problems  on  page  45  and  some  of  the  formal 
problems  on  page  46. 

Page  43:  13.  48,  132  IS-  76I,  103^ 


14-  41,  139 


Page  44:  16.  35,  145 


17.  79I,  ioo|;  7if,  io8i;  53I,  126I; 


2 


i8o+^f 


2 


18.  71J,  io8| 

19.  30, 221, 40 

20.  36,  20,  16/1-,  Sif)  I02f,  160 


Page  45:  23.  80,  100 


24.  so,  130 
25-  45 


26.  40,  140 

27.  12,  168 


28.  :r=iis 
69,  III 

29.  ^i;:=252 
182,  “2 

30.  X=I26 
II9,  61 


22 


First-Year  Mathematics  Manual 


[pP-45- 


31- 

2:  = S4 

37- 

x = 6o 

^79,  I 

68,  112 

32- 

X=2^ 

38. 

X=22.4 

105,  75 

91 . 2,  8^8 

33- 

a:=24 

39- 

00 

II 

55,  125 

5°1 7, 

34- 

x=45 

1 29  A 

96,  84 

35- 

40. 

x = go 

a:  = 30 

97, 83 

27,  153 

36. 

x = 90 

41. 

X=24 

153,  27 

55,  125. 

Lesson  5 : pages  46-50 

Let  the  class  read  rapidly  or  the  teacher  may  present 
the  subject-matter  of  §§36  and  37  to  Problem  4.  Let 
the  class  work  orally  the  easier  problems  on  page  49  and  50. 
Assign  for  home  work  Problem  4 and  some  of  the  more  diffi- 
cult problems  on  page  49,  asking  for  drawings  for  some  of 
them  only.  Assign  for  home  work  some  of  the  exercises  on 
page  so. 


Exercise  VI:  page  46 


I.  115 

7- 

4 

13- 

7 

2.  14 

8. 

2 

14. 

3 

4 

3-  6 

9- 

4 

IS- 

10 

4.  20 

10. 

4 

16. 

5 

5-  10 

II. 

7 

17- 

I. 

6.  5 

12. 

I 

Page  49: 

6. 

X=20 

167 

8. 

x = 40 
40? 

10.  x = 

168 

84 

7- 

x = 6o 
163 

9- 

2^=55 

117 

II 

M 

M 

168 

-5-f  ] The  Equation  Applied  to  Angles  23 


12. 

0 

II 

17- 

*=56 

22. 

*=55 

230 

48,  132 

117, 63 

13- 

11 

18. 

* = 4S 

23- 

*=45 

114 

33,  147 

33, 147 

14. 

*=72,36 

19. 

25=36 

24. 

*=45x1 

153,  27 

i37tV, 

IS- 

x = 4,  II 

20. 

X = ()0 

42H 

260,  —80 

25- 

2=36 

16. 

x=io 

21. 

x=6o 

27,  153- 

i79i  1 

13s,  45 

Lesson  6:  Pages  50-54 

This  lesson  is  very  much  like  the  preceding  one.  Part 
of  the  time  may  be  used  to  review  rapidly  the  essential 
points  of  Lesson  2.  For  home  work  assign  some  of  the 
problems  on  page  50  which  were  not  taken  up  in  Lesson  6, 
and  some  problems  on  pages  52,  53,  and  54. 


Exercise  VII 


I. 

2 

9- 

I 

17- 

42 

2. 

2 

10. 

identity 

18. 

IS 

3- 

3 

II. 

7 

19. 

21 

4- 

9 

12. 

5 

20. 

8 

5- 

7 

13- 

14 

21. 

8 

6. 

9 

14. 

3 

22. 

7- 

3 

IS- 

8 

8. 

10 

16. 

II 

Pages  51 

, 52,  53,  54: 

5- 

30°, 

60°,  79f,° 

44f°,  (90-0)°,  *° 

6. 

(90- 

-n)° 

7- 

(90- 

-d)°,  (90- 

-3C)°, 

1 

T 

0 

^ 0 

0 

8.  50" 


24 


First-Year  Mathematics  Manual  [pp,  51- 


o o 

9.  70 , 20 
10.  d°,c° 

13.  21,  69 

14-  33,  57 

15-  59,  31 

16.  35,  865 

17-  34i,  SSi;  26f,  63^;  8f, 

18.  58i  31^ 

19.  i8°,-9,f°,  W° 

20.  Y°,  -¥r,  50°,  81° 

23.  30°,  60° 

24.  40°,  50° 

25.  60°,  30 

26.  45,  45 

27.  12,  78 


2 


Lesson  7 : 54  to  the  end  of  the  chapter 

Let  the  class  work  out  Problem  i,  page  54,  with  some 
pupil  making  the  drawing  on  the  board.  This  should 
bring  out  a statement  of  Theorem  I,  page  56.  Problems  2,  4, 
and  6 all  bring  out  the  same  principle  and  can  be  worked 
in  very  little  time  by  the  teacher  and  class  together.  Prob- 
lem 7,  page  55,  is  important  as  it  leads  to  an  algebraic  state- 
ment of  the  relation  between  the  interior  angles  of  a triangle. 
All  of  §§  39,  40,  and  Problem  i of  §41  may  be  developed 
by  the  teacher. 

Let  the  class  work  some  of  the  problems  on  page  57,  one 
or  two  of  the  parts  of  Problem  19,  and  some  of  the  problems 
on  page  59. 

Assign  for  home  work  some  problems  on  pages  57,  58,  and 
59  and  the  summary  at  the  end  of  the  chapter.  All  theorems 
and  definitions  should  be  well  learned. 


The  Equation  Applied  to  Angles 


25 


-59] 


Pages  56,  57,  58: 

2.  54,  18,  108 

3.  40,  20,  120 

4.  108,  18,  54 

6.  40,  120,  20 

7.  76,  58,  46 

8.  62,  82 

9.  41,  16,  123 

10.  42,  21,  117 

11.  31I,  126,  22^ 

12.  132,  40,  8 

13.  1 14,  16,  50 

17.  w=io8 
54,  36 

18.  45 

19.  114/r,  32 A,  54,  36,  18,  108,  54,  30,  120 

20.  60 

21.  120,  60 

Exercise  VIII:  page  58 


I.  ¥ 

3- 

I 

5-  3 

2.  I 

4- 

5 

6.  10 

Page  59: 

7- 

6 

IS- 

5 

23.  a 

8. 

6 

16. 

10 

24.  2a 

9- 

15 

17- 

I 

25.  6a 

10. 

56 

18. 

I 

^ 12b 

26.  

II. 

6 

19. 

4 

5 

12. 

I 

20. 

I 

27. 

13- 

7 

21. 

a 

28. 

14. 

2 

22. 

2+sa 

2 


[pp.  62- 


CHAPTER  IV 

POSITIVE  AND  NEGATIVE  NUMBERS 

If  the  work  which  precedes  this  chapter  has  been  well 
done  the  pupil  has  sensed  sufficiently  well  for  a beginner 
the  value  of  the  equation  as  a tool  for  stating  and  solving 
problems.  He  has  also  reviewed,  under  the  guise  of  algebra, 
the  essentials  of  arithmetic,  and  has  generalized  many  of  the 
laws  of  arithmetic,  through  the  agency  of  literal  number, 
into  algebraic  formulas.  Furthermore,  he  has  put  into 
algebraic  garb  the  mensurational  laws  for  areas  and  for 
angles.  He  has  thus  seen  that  even  in  the  field  of  applied 
numbers,  already  partially  at  his  command,  algebraic  formulas 
and  equations  are  compact  and  convenient  short-hand 
modes  of  expressing  and  recording  laws  of  number  and  of 
magnitude,  and  that  the  negative  number,  or  something 
equivalent  to  it,  is  demanded  even  in  ordinary  problem- 
solving. 

The  next  natural  step  is  to  learn  more  fully  what  a nega- 
tive number  is,  what  the  extended  field  of  positive  and  nega- 
tive number  is,  what  addition,  subtraction,  multiplication, 
and  division  of  these  new  numbers  mean,  and  how  these 
operations  are  to  be  applied  to  these  new  numbers.  All 
this  is  done  most  effectively  by  means  of  easy  real  problems 
and  exercises  that  employ  the  positive-negative  notion. 

It  is  recommended  that  this  chapter  be  covered  in  7 
lessons  at  most.  The  material  may  well  be  divided  into 
lessons  somewhat  as  follows: 

First  lesson:  to  § 46,  page  67. 

Second  lesson:  to  Problem  8,  page  71. 

Third  lesson:  to  Problem  3,  bottom  page  73. 

26 


-64] 


Positive  and  Negative  Numbers 


27 


Fourth  lesson:  to  Multiplying  Positive  and  Negative 
Numbers,  page  77. 

Fifth  lesson:  to  Dividing  Positive  and  Negative  Numbers, 
page  85. 

Sixth  lesson:  to  Summary,  page  88. 

Seventh  lesson:  Study  the  Summary  and  review  the  chap- 
ter. 

§ 42.  Let  the  teacher,  or  better,  let  some  member  of  the 
class  read  Exercises  i,  2,  3,  4,  and  have  different  members 
of  the  class  answer  the  questions  or  give  the  results  of  the 
exercises  orally  and  rapidly,  that  the  connection  and  logical 
significance  of  the  individual  exercises  may  be  felt.  Have 
the  answers  of  Exercise  3 come  right  along  in  the  abbrevia- 
tions, thus: 


(5)  Rix-sT 

(6)  R(a-b)° 


3.  (i)  R 13° 
(2)  R3° 


(3)  R4° 

(4)  F 5° 


If  on  R(:r— 5)°  of  (5)  and  on  R(a— 6)°  of  (6)  the  question 
arises  whether  they  should  be  R(5— x)°  and  R(6— clear 
the  matter  up  for  the  class. 


4.  (i)  +12^ 

(2)  +8° 

(3)  +2° 

(4)  -3° 


(5)  (^+y)  degrees 

(6)  (a—x)  degrees 

(7)  a—a,oro  degrees 

(8)  (—a—x),  or  —(a+x)  degrees 


5.  Have  pupils  show  points  on  the  graph  located  by  the 
readings: 


The  teacher  may  study  the  graph  in  the  book  with  the 
class,  and  then  with  its  help  he  may  put  the  graph  of  one  of 
the  subsequent  problems  on  the  board.  This  will  save  time, 
increase  the  class  interest,  and  bring  up  new  difficulties  for 
immediate  explanation,  or  the  teacher  may  prefer  to  transfer 
this  figure,  or  to  draw  before  the  eyes  of  the  class  a similar 
one  on  the  cross-lined  blackboard,  and  thus  to  make  clear 


28 


First-Year  Mathematics  Manual  [pp.  6j- 


the  location  of  the  points,  and  the  meaning  of  the  broken 
connecting  line. 

The  answers  to  the  questions  in  the  paragraph  below 
Fig.  67  should  be:  it  rose;  it  rose  slowly;  rose  rapidly;  rose 
slowly;  stood  still;  fell  rapidly;  fell  rapidly;  fell  rapidly; 
first  4 hours j a rise;  5th  hour,  stationary;  last  3 hours,  a fall. 
Dwell  on  this  work  only  long  enough  to  get  the  major  ideas 
of  it  before  the  class.  Thoroughness  is  impossible  at  this 
stage  of  advance  and  it  is  a pedagogical  mistake,  bordering 
on  pedanticism,  to  spend  much  time  in  attempting  to  be 
thorough  with  elementary  and  poorly  defined  notions.  If  the 
work  cannot  all  be  done  in  one  class  period  assign  some  of 
it  for  home  work.  This  has  the  added  merit  of  giving  the 
pupil  a chance  to  review  a little. 

6.  Let  pupils  use  notebooks  supplied  with  cross-lined 
pages.  When  all  have  tried  and  some  are  through,  let  the 
teacher  quickly  lay  off  these  readings  on  the  cross-ruled 
blackboard  and  draw  the  broken  connecting  line,  as  a 
resume. 

7.  Move  along  rapidly  with  the  plotting  here.  Do  not 
allow  ‘‘puttering.” 

§43.  Have  3 or  4 pupils  solve  Exercise  i,  page  64,  and 
have  one  pupil  explain  it  to  the  class.  Pass  briskly  along 
through  Exercises  2,  3,  4,  and  5.  Tarry  long  enough  on 
6 to  allow  pupils  to  sense  the  nature  of  the  forces,  then 
require  the  answers  to  come  rapidly. 


-66] 


Positive  and  Negative  Numbers 


29 


From  here  on  push  briskly  through  the  list  of  problems 
to  §44.  See  that  the  answers  to  23,  to  be  done  orally,  are 
given  correctly. 


§ 43 — Answers 


2. 

-[-18  — io  = -t-8 

3- 

(i)  +2  mi. 

(3)  +150  mi- 

(2)  0 

(4)  {a-\-b)  mi. 

4- 

(i)  +10  mi. 

(4)  a+c  mi. 

(2)  +10  mi. 

(5) 

m—n  mi. 

(3)  3-wmi. 

(6) 

n—m  mi. 

S- 

J ton  pulling  forward 

6. 

(i)  F 8 oz. 

(3)  F 10  tons 

(2)  F 8 lb. 

(4)  F 15  tons 

7* 

(i)  +8 

(4)  -7 

(7) 

(2)  0 

(s)  -24 

(8)  a—b 

(3)  -23 

(6)  x—12 

1 

<3 

1 

8. 

Rise  with  3 oz. 

force 

9- 

(i)  +10  lb. 

(3)  -33  lb- 

(5)  ^+y  lb- 

(2)  +7  lb. 

(4)  -13  lb. 

(6)  0 

10. 

(i)  30° 

(4)  -12" 

(7)  x°-y° 

(2)  -16° 

(S)  ^°-io° 

(8) 

(3)  0° 

(6)  -2;°- 10° 

II. 

IOC. 

12. 

P $200,  P $23. 

, D $15,  P $700 

13* 

8 lb.  upward 

14. 

. 85  years 

IS- 

(i)  IS 

(3)  2 

(s)  300 

(2)  70 

(4)  150 

(6)  300 

16. 

51  years 

17- 

23  years 

30 


First-Year  Mathematics  Manual  [pp.  66- 


18. 

646  years 

19. 

64  years 

20. 

— 60  rd.,  —i\  mi. 

1 

+ 

00 

0 

21. 

6 mi.  northward. 

+6  mi. 

22. 

+ 9 mi.  — 9 mi. 

23- 

(i)  below 

(4)  leftward 

(2)  backward 

(5)  before 

(3)  downward 

(6)  west 

(7)  south 

(8)  debts. 


Have  different  pupils  read  the  several  paragraphs  of  page 
67,  the  teacher  noting  when  the  reading  indicates  that  the 
ideas  are  being  comprehended  by  the  reader  and  by  the  class, 
or  to  save  time  the  teacher  may  first,  present  these  ideas  to 
the  class  and  then  refer  to  page  67  for  re-reading.  Proceed 
thus  to  the  phrase  Graphing  Data,  page  68. 

§42.  Have  3 or  4 pupils  make  the  graph  of  Exercise  i, 
page  68,  and  bring  it  to  class  next  day;  3 or  4 others  make 
the  graph  of  Exercise  2,  and  so  on  through  the  list  to  page 
70.  It  is  perhaps  still  better  to  assign  a little  of  this  graphic 
work  from  day  to  day  for  several  days,  than  to  condense  it 
all  into  one  day. 


Graphing  Precise  Laws 

Exercise  i. — First  have  pupils  calculate  the  areas  of 
the  rectangles  from  the  given  data  and  record  the  areas  beside 
the  given  lengths  in  tabular  form  in  their  notebooks.  Then 
have  pupils  take  cross-ruled  paper  and  plot  each  base-length 
and  its  corresponding  area  and  mark  the  points  clearly  and 
neatly  and  then  draw  the  connecting  line.  Before  this  work 
is  forgotten,  have  Exercise  2 answered.  Notice  that  the 
intention  is  that  the  pictured  form  shall  precede  the  equa- 
tional  form,  y = 3X.  The  thought  is  that  the  equational 


-72] 


Positive  and  Negative  Numbers 


31 


form  should  be  seized  by  the  learner  as  the  short-hand  de- 
scription of  the  pictured  law. 

3.  If  Exercise  3 is  not  answered  at  once,  don’t  tell,  but 
ask  that  Exercise  2 be  reanswered,  and  then  try  Exercise  3. 

4.  Let  the  pupils  answer  Exercise  4 without  help  if 
they  can.  If  they  cannot,  the  difficulty  is  likely  to  be  with 
the  question  “ state  how  the  area,  y,  varies  with  the  altitude,  ” 
etc.  Let  the  teacher  change  the  question  to  “If  the  altitude 
is  doubled  how  is  the  area  changed?”  “If  trebled,”  etc. 

In  the  same  fashion  work  down  through  the  Exercises 
of  page  71. 

§ 49.  Have  each  pupil  graph  at  least  3 of  the  Exercises 
1-9. 

§ 50.  Have  each  pupil  graph  in  class,  or  as  home  work. 
Exercise  i and  at  least  3 parts  of  Exercise  2.  Have  all 
the  graphs  of  the  parts  of  Exercise  2 made  on  the  same 
drawing.  Here  again  it  is  suggested  that  a little  of  this 
work  from  day  to  day  for  several  days  will  give  better 
results  than  to  complete  it  all  at  once. 


Adding  Positive  and  Negative  Numbers 

Work  orally  with  the  class  through  all  the  parts  of 
Exercises  i,  2,  3,  and  4,  answering,  or  prompting,  or  cor- 
recting only  where  you  feel  it  necessary  to  prevent  the  work 
from  dragging,  from  wool-gathering,  and  the  point  of  the 
development  from  being  lost. 

The  idea  is  that  the  pupil  shall  put  himself  as  fully  as  is 
practicable  into  the  work  of  developing  the  addition  laws. 
Telling  will  be  fatal  to  interest  here.  If  only  a few  of  the 
exercises  of  2 are  really  needed  to  develop  the  rule  the  rest 
may  be  omitted,  though  the  teacher  is  in  danger  of  using 
too  few,  with  the  thought  that  they  are  to  function  only  as 
illustrations. 


32 


First-  Y ear  M aihematics  M anual  [pp.  72- 


Answers  to  exercises: 


I.  (i)  +5  mi. 

(6)  —22  mi. 

(2)  +i  mi. 

(7)  -5  mi. 

(3)  -5  mi. 

(8)  +7  mi. 

(4)  —10  mi. 

(9)  +i  mi. 

(5)  +6  mi. 

(10)  0 

2.  (i)  +23 

(6)  -57 

(ii)  -15 

(2)  -23 

(7)  +19 

(12)  +22 

(3)  +7 

(8)  —19 

(13)  0 

(4)  -7 

(9)  +30 

(S)  +57 

(10)  -30 

3.  Algebraic  sum  is  arithmetical  sum  with  common  sign 
prefixed. 

4.  Algebraic  sum  is  arithmetical  difference  with  sign  of 
larger  prefixed. 

§ 52.  The  thought  with  Exercise  i is  that  the  pupil 
shall  use  any  common-sense  way  of  combining  the  numbers 
into  a sum  that  may  occur  to  him  after  having  learned  how  to 
proceed  to  find  the  sum  of  two  signed  numbers.  In  answer- 
ing Exercise  2 whether  he  give  either  of  the  two  ways 
stated  in  paragraph  5 of  the  Summary,  page  88,  or  some  other 
plan  of  his  own  is  immaterial  just  here.  A real  effort  is  all 
that  is  needed  to  secure  concentration  upon  the  idea. 

Have  as  many  of  the  examples  from  Exercise  3,  page 
73,  to  the  bottom  of  page  75  solved  orally  as  the  pupils  can 
do  comfortably. 

Answers  to  exercises: 

1.  (i)  +51  (4)  +10  (7)  +i2a 

(2)  +6  (5)  +494 

(3)  +50  (6)  +5^ 

2.  (i)  Add  the  several  numbers  in  order,  or  (2)  add  all 

positives,  then  all  negatives,  then  add  the  two  sums. 


-75] 


Positive  and  Negative  Numbers 


33 


3.  3 lb.  toward  right 


4.  (i)  +4  lb. 

(2)  -4  lb. 

(3)  o 

(4)  -3  lb. 

(5)  -3  lb. 

(6)  —II  lb. 


(7)  +26  lb. 

(8)  -4  lb. 

(9)  -241b. 

(10)  ~9  lb. 

(11)  x-\-y  lb. 

(12)  x—y  lb. 


5.  5 lb.  upward 

6.  2 oz.  upward 

7.  i5f  lb.  downward 

8.  78  feet  above,  or  +78  feet 
9-  +3° 

10.  5 mi.,  I mi. 


(14)  -x+y\h. 

(15)  o 

(16)  — xlb. 


Exercise  IX 


I. 

+if 

13- 

+9I 

2. 

1 

8 

14. 

+I-3S 

3- 

IS- 

-2.3 

4- 

+i| 

16. 

+ 2r 

5- 

-7f 

17- 

’-lOS 

6. 

-If 

18. 

d-fx 

7* 

+ 2f 

19. 

+46*^ 

8. 

20. 

9* 

-IS 

21. 

-\-2{x—a) 

10. 

+7li 

22. 

— 19  {x+y) 

II. 

-4 

23- 

-143  {m-r) 

12. 

-.92 

24. 

-2f  {c-d) 

Exercise  IX,  page  75,  particularly  should  be  done 
orally  as  far  as  possible,  that  the  pupil  may  get  the  training 
in  mental  arithmetic  with  simple  fractions  and  that  he  may 
feel  that  two  things  must  be  paid  attention  to  in  obtaining 
algebraic  sums,  viz.,  the  absolute  value  and  the  algebraic 
sign. 


34 


First-Year  Mathematics  Manual  [pp.  75- 


Solve  in  the  class  the  exercises  that  follow,  to  the  title 
^^Multiplying  Positive  and  Negative  Numbers,^  page  77.  Do 
not  solve  the  exercises  for  the  class  but  with  the  class,  the 
class  doing  the  work.  In  Problems  i and  2,  at  bottom  of 
page  77,  write  the  results  in  the  book  for  later  use. 

Make  Exercise  X as  far  as  possible  an  oral  class  exercise. 

Read  and  call  for  oral  answers  to  the  questions  of  Exer- 
cises 1-12,  pages'  77-78.  The  purpose  of  these  exercises 
is  to  start  the  ideas  of  multiplication  with  a concrete  multi- 
plicand and  an  unsigned  multiplier.  Let  pupils  do  the 
answering. 

§ 55.  The  experimental  exercises  of  page  79  should  not 
be  too  long  dwelt  upon.  The  thought  is  that  the  turning 
bar  furnishes  an  objective  situation  for  the  multiplicative 
use  of  the  positive  and  negative  signs  to  describe.  The  pupil 
needs  only  to  see  that  following  the  ordinary  law  of  multi- 
plication adequately  describes  the  behavior  of  the  loaded 
bar.  The  work  should  be  conducted  orally  and,  if  possible, 
with  an  apparatus  before  the  class. 

§§  58-59-60  are  to  make  more  explicit  the  mathematical 
ideas  exemplified  in  §§  55-56-57. 

§§  61-62,  to  be  treated  orally,  are  to  give  a certain  hold 
on  the  mathematics  of  turning-tendencies. 

§ 63  furnishes  a straight-line  interpretation  of  the  multi- 
plication of  signed  numbers.  It  is  not  superfluous  to  attempt 
to  underlay  the  laws  of  signed  numbers  with  a foundation  of 
clear  ideas. 

The  ideas  that  are  informally  used  through  the  five 
preceding  exercises  and  exemplified  in  the  experimental 
exercises  with  the  turning  bar  are  summarized  into  the  laws 
of  multiplication,  first  ‘‘in  digits’’  and  then  “in  letters.” 
After  the  pupil  has  felt  that  these  laws  have  a high  descrip- 
tive value  as  records  of  the  real  behavior  of  the  balanced 
bar,  and  has  sensed  their  simple  use  in  directed  measure- 


-86] 


Positive  and  Negative  Numbers 


35 


ments  along  a line,  the  general  a6-formulation  of  them  will 
seem  to  him  plausible,  at  least;  and  the  attitude  of  mind 
of  the  learner  toward  them  will  be  much  more  serious  and 
more  rational  than  if  he  first  be  given  those  laws  as  mere 
‘^definitions  of  multiplication.” 

Exercise  6 first  focuses  the  pupil’s  mind  on  the  ques- 
tion of  obtaining  a rule  of  practice  for  multiplication.  Then 
after  the  pupil  has  tried  by  his  own  efforts  to  formulate 
such  a rule,  §64  gives  him  a standard  of  excellence  for 
getting  this  rule  into  finished  form.  This  procedure,  though 
a little  time-consuming  in  the  initial  stages,  more  than 
remunerates  in  rapidity  and  steadiness  of  progress  afterward. 

Again,  only  part  of  the  exercises  under  3,  page  84,  may 
be  needed.  Pupils  should,  however,  make  drawings  and 
obtain  the  results  from  the  drawings,  not  from  a memorized 
rule. 

Exercise  XI  immediately  applies  the  rule  just  made  and 
gives  a review  of  the  arithmetic  of  fractional  and  mixed 
numbers  under  the  appearance  of  algebra. 


Dividing  Positive  and  Negative  Numbers 

§65.  Division  may  now  be  given  a direct  treatment 
by  basing  it  upon  multiplication  at  once.  Exercises  i and 
2 are  to  steady  the  pupil’s  first  steps  in  using  the  division 
law  by  exhibiting  the  law  through  the  simple  numbers  of 
arithmetic,  that  he  may  give  his  undivided  attention  to  the 
sign-phase  of  the  law,  and  Exercise  3 then  generalizes  to 
the  use  of  this  law  with  literal  numbers. 

Exercise  4 is  intended  to  be  an  oral  exercise  to  give  a 
little  practice  and  a modicum  of  steadiness  in  answering 
questions  on  the  basis  of  the  division-law,  and  Exercise  5 
is  to  aid  the  pupil  toward  a statement  of  the  law  of  signs 
for  division.  Exercise  6 requires  him  to  attempt  the 


36 


First-Year  Mathematics  Manual 


[pp.  86- 


statement  for  himself,  and  finally  § 67  gives  him  a standard 
for  perfecting  his  own  statement.  The  exercises  that  follow 
immediately  put  this  law  into  use.  Exercise  XII  at  once 
applies  the  division  law  and  reviews  division  of  arithmetical 
fractions.  Have  as  many  as  possible  of  these  exercises  done 
orally. 

Let  the  Summary,  page  88,  now  be  assigned  for  home 
work,  the  class  being  told  to  learn  the  definitions  of  graphs, 
of  algebraic  sum,  algebraic  difference,  and  the  laws  for  multi- 
plication and  for  division,  and  to  be  ready  to  give  examples, 
both  with  arithmetical  and  with  literal  numbers,  to  illustrate 
all  of  them.  The  next  day  go  rapidly  over  the  results  of 
these  attempts,  clear  away  any  confusion  that  remains,  and 
see  to  it  that  the  class  as  a whole  understands  these  defini- 
tions and  laws  very  fully. 


-8g] 


CHAPTER  V 

BEAM  PROBLEMS  IN  ONE  OR  TWO  UNKNOWNS 
Problems  in  One  Unknown  Number 

§ 68.  Pupils  have  now  studied  rather  carefully  the  laws 
of  algebraic  addition,  subtraction,  multiplication,  and  divi- 
sion, and  have  applied  them  to  a considerable  number  of 
formal  problems  employing  both  arithmetical  and  literal 
numbers.  They  have  a satisfactory  first  hold  on  these  laws. 
It  is  but  fair  to  the  learner  that  he  should  have  an  oppor- 
tunity to  employ  these  new  possessions  on  some  sort  of 
practical  problems  before  he  grows  weary  of  a too  extended 
experience  with  purely  formal  exercises.  It  will  be  found 
to  be  a valuable  asset  to  the  teaching  of  algebra  to  enable 
the  pupil  to  bring  a little  of  his  high  regard  for  the  importance 
of  practical  matters  to  sanction  the  worth  of  his  algebraic 
study,  before  his  natural  disposition  to  discredit  empty 
forms  has  yet  been  developed.  Prevention  is  both  more 
economical  and  better  than  cure.  It  is  believed  that  the 
simple  exercises  of  this  chapter  will  open  a field  that  is  very 
rich  in  algebraic  demands  that  are  simple  and  real  enough 
to  enable  the  pupil  to  see  that  even  in  a practical  way  algebra 
is  highly  useful,  and  perhaps  worth  the  time  and  effort 
required  to  learn  it. 

It  is  intended  that  the  pupil  shall  work  through  this 
chapter  without  a knowledge  of  any  of  the  formal  methods 
of  solving  simultaneous  equations.  It  is  of  course  assumed 
that  he  can  transform  equations  with  some  intelligence  by  aid 
of  the  process  axioms  of  § 21,  page  26.  His  interpretations 
are,  however,  to  be  obtained  from  the  behavior  of  the  loaded 
bar.  The  idea  here  is  that  the  importance  of  the  equation 
as  a tool  for  problem-solving,  rather  than  formal  processes 

37 


38  First-Year  Mathematics  Manual  [pp- Sg- 

of  solving  equations,  shall  be  sensed.  Common-sense, 
visual  interpretations,  and  convictions  are  here  wanted. 
Formal  processes  come  later. 

This  chapter  may  be  well  assigned  in  accordance  with  the 
following  suggested  daily  subdivisions: 

First  lesson:  pages  89-94  inclusive. 

Second  lesson:  pages  92-94,  including  Problem  ii. 

Third  lesson:  page  94,  Problem  12,  to  page  97,  § 73. 

Fourth  lesson:  pages  97-102  inclusive. 

Fifth  lesson:  pages  103  and  104. 

Sixth  lesson:  pages  104  and  105  to  Summary. 

Seventh  lesson:  the  Summary  and  a light  resurvey  of 
the  chapter. 

The  plan  of  doing  some  of  the  problems  of  this  chapter 
with  work  of  the  next  chapter  is  good. 

Exercises  i and  2,  page  89,  and  the  statements  of 
§ 69  are  to  lead  up  to  the  statement  of  turning-tendency  as 
an  algebraic  sum,  which  is  given  in  § 70. 

Calling  the  turning-tendency  t,  have  the  several  answers 
to  the  parts  of  Exercise  i,  page  89,  written  thus  (obtaining 
the  sign  of  the  final  product  in  each  loading  by  raising  the 


question : In  which  direction  does  the  loaded  bar  turn  ?) 

1. 1 

= (+3)  (-6) 

= -i8 

II.  t 

= (+2)  (-3) 

= -6 

III.  t 

= (+3)  (+2) 

= +6 

IV.  t 

= W (+3)  = 

+32: 

V.  t 

= {x)  (-2)  = 

— 2X 

VI.  t 

= (+3)  W = 

+32: 

VII.  t 

= (-4)  (-9) 

= +36 

VIII.  / 

= (-3)  (-2) 

= +6 

IX.  t 

= (-12)  (+3)  = -36 

X.  1 

= (x)  (-3)  = 

-3^ 

XI.  t 

= {x)  (+2)  = 

+ 2X 

XII.  1 

= (-2)  (x)  = 

— 2X. 

-g2  ] Beam  Problems  in  One  or  Two  Unknowns 


39 


Exercises  1-3,  § 70,  are  intended  to  lead  to  the  final 
working  form  of  the  law  of  leverage  as  stated  in  § 71. 

Answers  to  Exercise  i,  § 70 

I-  (+3)  (-6)+(-i2)  (+3)  = -i8-36=-54 
2.  (+2)  (-3)+(-4)  (-9)  = -6+36  = 4-30 

3-  (+3)  (-6)+(+3)  (+2)+(-i2)  (+3)  = -i8+6-36 
= -48 

4-  (+3)(-6)  + (+3)  (H-2)  + (-4)  (-9)  = -18+6+36  = 
+ 24. 

Answers  to  Exercise  3,  page  91 

1.  3/-i8  = o,/=6 

II.  — 18  — 2ie^  = o,  — 9 

III.  -2r-3r+36  = o,  r = 7| 

IV.  — 2(/+3^/— 6 = 0,  J = 6 

V.  — 2/-t-3?“|"6  = o,  / = — 6. 

§71,  Exercise  i:  (+io)  (-6)+(+s)  (5+3)=o,  -6o+ 
55+15=0,  55=4S>  ^=9  and  5+3  = 12. 

Exercise  2:  —4  (3i£i— i5)+(+i2)  (+3)=o,  — i2w+ 
60+36  = 0,  i2ze;  = 96,  w = ^j  32c;— 15  = 9. 

Exercise  3 : 

I.  -3(w+5)+6o=o,  -3ie»-i5+6o=o,  w=i5,  w+ 
5 = 20 

n.  7(<-3)+(-0(-8)+(+i3)(-3)=o>  7/-2i+8i- 
39  = 0,  15^  = 60,  <=4,  <-3=1,  -^=-4 

III.  +3(wz— 5)— 60=0,  w— 5 — 20=0,  m = 25,  w— 5 = 20 

IV.  — 24+i2^--8^  = o,  4^  = 24,  k = 6,  3^  = 18,  4^  = 24 

V.  (+39l)(-4)  + (5^)(+2§)  + (-i3l)'(-4)=o,  -157 

+12^5+52^=0,  i2|5=io5,  255=210,  5=8|, 
55=42. 


Hid 


40 


First-Year  Mathematics  Manual  [pp.  g2- 


Practical  Applications 


1.  Taking  the  fulcrum,  F,  for  turning-point,  we  have 

(-i)(-i,8oo)+(+6)(-a;)  = o 
or  900  = 0,  x = iso  (lb.). 

2.  With  fulcrum,  F,  as  turning-point,  we  have: 

(-|)(-i,8oo)+(+6i)(-x)=o 
or  6|x-- 450  = 0 

\^-x— 450  = 0,  25x=i,8oo.  x = J2  (lb.). 

3.  Call  w the  unknown  weight,  and  take  turning-point  as 

before,  then  (— i^)(— 2e^)  + (+6)(  — 2oo)+o,  1,200  = 0, 

2,400  = 0,  w = 2,400, 

4.  Call  the  load  I,  then  as  above  — 
(-2)(-0+(+34)(-2o)=o,  2l-6So  = o,  l = S40  (lb.). 

5-  (“2)(-/)+(+34)(“68)=o,  2/-2,3i2=o,  /=i,is6 

(lb.). 

6.  (+1)  ( — 2,4oo)  + (+6)  (+w)=o,  6w=i,2oo,  m = 2oo 
(lb.). 

7.  (+2)(-966)  + (+i4)(+/)=o,/=i38  (lb.). 

8.  (+i)(-966)  + (+i3)(+/)=o,/=  74tV  (lb.). 

9.  i+d)  ( — 966)  + (i2+J)  (+24i|)=o,  2,898  — 966^?+ 
24\\d  = o,  724^^=2,898,  d = 4 (ft.). 

10.  (+2)(-2£;)  + (+i4)(+i4o)=o,  ze;=98o'(lb.). 


II.  By  putting  the  fulcrum  very  near  the  middle  of  the 
rail  and  weighing  the  end.  Show  from  (+^/)(—2,ooo)  + 
(i2+^/)(+6o)=o,  that  if  the  weight  is  just  a ton  and  the 
balances  read  just  60  lb.,  the  distance,  d,  of  the  fulcrum 
from  the  middle  is  ff  ft.  Show  that  if  either  the  weight  is 
more,  or  the  balances  read  less,  d must  be  smaller.  Calling 
W and  R the  weight  and  balance  reading  in  pounds,  show 
720 


that  d = 


W-R 


12.  (+2)(-36o)+(+s)(+/)=o.  /-144  OM, 
(-36o)+(+4)(+/)  = o,/-9o  (lb,). 


(+1) 


-g8]  Beam  Problems  in  One  or  Two  Unknowns 


41 


13.  (+2)(-27o)  + (+4V)(+/)=0,  /=I20  (lb.). 

14.  (+J)(-27o)+(4§)(+9o)=o,  -2,d+A\=o,  d=i\ 

(ft.). 

Exercise  XIII:  Answers 


1.  -45-3^ 

2.  -850+85^ 

3.  2ll—^U 

4.  8^w-s4 
S-  90+42^ 

6.  5* 

7-  5^-7 


-5^+7 

o 


10.  o 

11.  o 

12.  O 

13' 

14 


— 2kl—2kx 


15.  15:^ 

16.  —gy 

17.  1SX-7 

18.  -15X+7 

19.  o 

20.  o. 


—gx 


§72.  I.  (+52:)(-4)+(-s6)(-i)+(+4)(+3^)=o,-202: 
+S6+i2x=o,  82:=s6,  x=’j,  5^=35,  32;=2i. 

Read  the  first  half  of  page  96  with  the  class. 

2.  (i)  (+5^)(-6)+(-9o)(+2)+(+65)(+6)=o, 


65  = 180,  5=30,  etc. 

(2)  (+44)(~8)  + (— 6f0(— 3)  + (+2^)(+8)=o, 

/ = 9|,  etc. 

3.  Measuring  all  lever-arms  from  the  left  end, 

(1)  (+3)(+3/)+(+i3)(-8oo)+(+I9)(+5/)  = o, 

104/=  104,00,  /=ioo,  3/=  300,  ands/=5oo. 
Also  (2)  (+1)  (+7w)+(+7)(-4s)+(+i2)(-6w)+ 

(+16)  (+40) =0,  7w— 3is-72w+640=o,  65^= 

325,  w=s,  etc. 

4.  (i)  (+3/)(~i8)+(— 8oo)(— 8)+(+5/)(— 2)=o, 

64/  = 6,400,  / = 100,  etc. 

(2)  (-i7)(+7w)+(-ii)(-4s)+(-6)(-6w)H- 
(-2)(+4o)=o,  -83^+415=0,  83^=415,  w= 


5,  etc. 


Problems  in  Two  Unknowns 

No  technique  is  wanted  here  save  the  two  laws  of  balance 
of  bars. 

Work  orally  and  carefully  with  the  class  through  page 
97  and  down  to  Exercise  7,  page  98. 


42 


First-Year  Mathematics  Manual  [pp,  g8- 


Exercise  7 : Have  the  class  substitute  from  half  a dozen 
to  a dozen  values  of  S in  equation  (2)  and  calculate  the 
corresponding  values  of  R.  Arrange  these  pairs  of  values 
and  plot  them.  Solve  Exercise  8 similarly. 

See  that  the  class  gets  the  point  of  Exercise  9. 

Exercise  10:  The  equations  of  this  exercise  are  to  be 
solved  by  plotting  and  not  by  algebra.  The  same  is  true  of 
Exercise  ii. 

Exercise  12:  Using  the  middle  point  as  turning-point, 
we  have 

(+x)(-4)  + (-84)(-2)  + (+)/)(+4)=o,  or  -x+y=42 
and  x-\-y  = ^^  is  easily  seen  to  be  true  from  the  problem. 

Solve  the  equations  by  graphing. 

§ 75.  Read  Exercises  1,2,  and  3,  and  solve  2 and  3 with 
the  class,  and  then  require  the  class  to  solve  Exercise  4. 

§ 76.  Have  the  class  read  and  explain  the  text  of  Exer- 
cises I and  2 of  this  section.  Work  the  remaining  exercises 
through  with  the  class,  not  telling  more  than  is  necessary. 

§ 77.  Be  sure  that  the  two  italicized  laws  of  this  section 
are  clearly  comprehended. 

Problems  Applying  Two  Unknowns 

Exercise  i:  Taking  the  turning-point  at  the  left  end, 
(+2)(+E)  + (+6)(-6)  + (+8)(+R)=o,  or  F+4R=i8 
from  law  of  leverages,  and  +F— 6+i2  = o,  or  E+R  = 6. 
Subtract  the  second  from  the  first  equation  and  obtain 
3^  = 12,  or  R = 4,  and  since  E+A!  = 6,  E = 2. 

Check:  2+4*4=18,  and  4+2  = 6.  Turning-point  might 
be  taken  at  either  the  right  end  or  the  middle. 

2.  Taking  the  turning-point  at  the  left  end  (Fig.  102)  and 
noting  that  z£;  = 3|X4X  18X48  = 12,096. 

From  the  law  of  leverages:  (+3)(+E)  + (+9)  ( — 12,096) 
+ (+12)  {R)=o,  or 


^+4/^  = 36,288 


(i) 


-10 j]  Beam  Problems  in  One  or  Two  Unknowns 


43 


and  from  the  law  of  forces  (F— 12,096+7^  = 0), 

7^+7^=12,096  (2) 

Subtract  (2)  from  (i)  37^  = 24,192,  or  7^  = 8,064,  and  from 
(2)  /?  = 4,032. 

Taking  the  turning-point  at  right  end,  with  2e;  = 12,096; 
Law  of  leverages,  (-i5)(+F)  + (-9)(-i2 ,096)  + ( - 6)  (+7?) 
= 0,  or 

5^+27^  = 36,288,  and  from  (i) 

Law  of  forces,  2^+27^  = 24,192  (2) 

Subtracting  (2)  from  (i)  37^=12,096,  or  7^  = 4,032  and 
from  F+7?=  12,096,  7^  = 8,064,  as  before. 

Taking  the  turning-point  at  the  middle,  2£;=  12,096; 
Law  of  leverages,  (-6)  (+7^)  + (o)  (-i2,o96)  + (+3)  (+7?) 
= 0,  or 

— 2F+7^  = o,  and  from  (i) 

Law  of  forces  as  before, 

7^+7^=12,096  (2) 

and  subtracting  (2)  from  (i),  37^=12,096,  or  ^ = 4,032,  and 
from  (2)  7^  = 8,064  as  before. 

Clearly  the  turning-point  may  be  taken  at  any  one  of  the 
three  points,  but  the  same  turning-point  must  be  used  through- 
out the  solution. 

3.  2e;  = 40X60  = 2,400. 

Taking  the  turning-point  at  the  left  end, 

(+i|)(+^)  + (+5)(”2,4oo)  + (+7J)(+7^)  =0,  or 

(1)  F+s7?  = 8,ooo,  and  F— 2,400+7^  = 0. 

(2)  F+7^  = 2,400,  and  subtracting  (2)  from  (i),  = 

5,600,  or  7^=1,400,  and  from  (2)  F=  1,000. 

This  exercise  also  may  be  solved  by  taking  the  turning- 
point  at  either  the  right  end  or  at  the  middle,  as  was  done 
above. 

4.  From  the  law  of  leverages  (+^)(  — s)  + (--i7o) 
(~3)+(— 3o)(o)  + (+3')(+5)  = o,  or 

x—y=  102 


(i) 


44 


First-Year  Mathematics  Manual 


[pp.  103- 


From  law  of  forces  (+^— 170— 3o+>'  = o) 

x-]ry  = 2oo  (2) 

whence  adding  (i)  and  (2)  2:r  = 302,  and  ^=151,  and  y = A9- 

5.  Turning-point  at  middle  of  span  of  bridge. 

(+^)  (”"7i)  + (“45o)  (“2|)  + (— 1,000)  (o)  + (7^)  (+7i)=o. 

Multiplying  and  simplifying,  L—R=i^o,  and  from 
law  of  forces,  +Z  — 1,450+7^  = 0,  or  Z+i^  = 1,450,  and 
adding,  2Z  = 1,600,  or  Z = 8oo. 

From  X+7^  = 1,450  and  L = 8oo,  R = 65o. 

6.  Turning-point  at  middle  of  span, 

(+L)  (-io)+(-6oo)  (-6)+(-2,4oo)  (o)+(-8oo)  (+5) 
+ (+i?)(+io)  =0,  or 

(1)  R— L = 4o,  and  from  Z— 600  — 2,400  — 800— i?  = o. 

(2)  R+Z  = 3,8oo. 

From  (i)  and  (2)  2^  = 3,840,  72  = 1,920,  and  Z=  1,880. 

7.  Using  middle  point  as  turning-point, 

(+L)  ( — io)  + (— 45o)(— 8)  + (— 45o)(— 4)  + (— 45o)(  — i) 

+ ( — 2,400)  (o)  + ( — 450)  (+6)  + (+72)  (+ 10)  = o, 
or  simplifying, 

(1)  Z— 72  = 315,  and  from  Z— 450— 450— 450— 2,400  — 
450+72  = 0. 

(2)  Z+72  = 4,2oo;  adding  2Z  = 4,425,  Z = 2,2I2|,  and  from 
(2)  72  = 1,942^ 

8.  From  the  law  of  forces  we  have  at  once  (i)  Z+72  = 
7,500,  and  from  the  law  of  leverages, 

(+Z)(-io)  + (-2,5oo)(-6)  + (-5,ooo)(o)  + (+72)(+io)=o 
or  simplifying  (2)  72— Z = 1,500. 

Equations  (i)  and  (2)  give  272  = 9,ooo,  72  = 4,500,  and 
Z = 3,000. 

9.  From  the  law  of  forces  (i)  72+Z  = 7,500,  and  from 
law  of  leverages,  turning-point  at  middle: 

(+Z)(-io)  + (-2,5oo)(-i)  + (-2,ooo)(o)  + (-3,ooo)(+5) 
+ (+72)(+io)=o 


-io6]  Beam  Problems  in  One  or  Two  Unknowns 


45 


or  (2)  — 1,250.  From  (i)  and  (2)  7^  = 4,375  and 

^ = 3,125. 

10.  From  the  law  of  forces  we  have  at  once  — 75  = 

o,  or  w=ioo,  and  law  of  leverages  gives  (taking  turning-point 
at  middle  and  calling  d the  distance  from  middle  to  load) : 

(+75)(-4)  + (-w)(-J)+(+2s)(+4)=o 

or  —300+2^;^+ 1 00  = 0,  or  wd  = 200,  But  2£;  = ioo.  Hence 
d=+2]  that  is,  the  load  hangs  2 ft.  from  the  middle  toward 
the  man  lifting  75  lb. 

11.  (+2/)(+{/)  + ( — 24o)(+6)  + (+/)(+i2)=o, 
or  2/J+i2/=  1,440. 

Also  +2/— 24o+/=o,  from  law  of  forces,  or  37=240,  and 
/=8o.  Putting  this  / in  the  equation 
2/J+I2/=I,440 

and  simplifying,  ^+6  = 9,  or  c/  = 3. 

The  spike  should  be  placed  3 ft.  from  the  end  of  the  log. 

12.  First  equation  (+2/)  (o)+(  — 24o)(6—J)  + (+/)  (12  — 
J)  =0,  where /=8o  as  before. 

Dropping  (+2/)(o)  and  dividing  through  by  80, 
(-3)(6-<^)+(i2-ii)=o 

or  — i8+3(i+i2— 6=0,  or  2d=6,  d=^,  as  before. 


Exercise  XIV 

(1)  5=8,  i=6  (5)  t=7,w=5 

(2) 7=2,  w=5  (6)  x=4,y=8 

(3)  k = 4,l  = 3 (7) 

(4)  x=so,  w=4  (8)  x=8^r,  y=iSlf 

Require  pupils  to  learn  the  substance  of  the  statements 

3,  4,  and  5 of  the  summary  very  thoroughly,  giving  more 
attention  to  securing  an  understanding  of  them  than  to 
memorizing  them  verbatim. 


[pp.  107- 


CHAPTER  VI 

PROBLEMS  IN  PROPORTION  AND  SIMILARITY 

Lesson  i : through  page  log 

There  should  be  cross-lines  painted  on  the  blackboard. 
If  not,  the  teacher  should  rule  such  lines  with  crayon  before 
the  recitation  hour.  Make  the  parallel  lines  i|  inch  apart 
and  rule  every  fifth  one  heavy. 

State  to  the  class  Problem  i.  Take  5 small  squares  to 
represent  each  yard  walked.  Locate  points  O and  M,  and 
draw  line  O M.  Ask  class  what  the  line  O M represents  in 
the  problem,  also  how  by  this  line  we  can  calculate  the 
number  of  miles  M is  distant  from  O.  Lay  a ruler  along 

0 M,  note  the  length.  Lay  the  ruler  horizontally  and  note 
the  number  of  squares  included  in  this  length  (37I).  As 
each  5 squares  stand  for  10  miles,  there  are  twice  as  many 
miles  as  small  squares,  i.e.,  the  37I  squares  represent  75 
miles. 

This  introduces  ratio  as  a mathematical  relation  and  a 
means  of  calculation.  The  relation  of  5 squares  to  10 
miles  is  called  the  ratio  of  5 to  10,  or  The  relation  of 

1 sq.  to  2 miles  is  the  ratio  of  i to  2 or  J,  and  these  ratios 
are  the  same,  i.e.,  the  number  of  squares  is  ^ the  number  of 
miles. 

Problem  2 : Have  the  pupils  notice  that  the  same  figure 
is  used  but  that  the  ratio  of  squares  to  miles  is  changed, 
5 small  squares  now  standing  for  15  miles.  Ask  what  the 
ratio  is  (i  to  3,  or  ^).  Every  square  now  represents  3 miles. 
Ask  how  many  miles  line  O A represents  (67^);  how  many 
A M represents  (90),  and  O M (ii2§). 

In  Problem  3 let  the  teacher  draw  in  class  on  the  board 
the  required  lines,  and  find  O M to  measure  12J  cm.  (or 

46 


-no]  Problems  in  Proportion  and  Similarity 


47 


large  squares),  more  exactly  i2iVo.  Have  class  tell  the 
ratio  of  cm.  to  yd.  (ya)  and  state  how  to  find  the  number  of 
miles  12^  cm.  represents  (150).  Assign  Problem  4 for  home 
work. 

Taking  i cm.  to  i mi.,  the  distance  apart  is  12.2  mi., 
approximately. 

Problems  5,  7,  8,  and  9 can  be  done  rapidly  in  class  as 
exercises  in  finding  the  scale  of  a drawing,  or  the  ratio  of  the 
line  unit  to  the  number  of  feet. 

Problem  6 shows  the  application  of  the  scale  to  making  the 
drawing,  and  can  be  given  as  home  work,  together  with 
Problems  9,  10,  ii. 

Page  109  furnishes  practice  in  methods  of  writing  ratios, 
and  in  their  meaning  and  use.  It  is  important  that  pupils 
get  a clear  understanding  of  these  things. 

§ 79  may  be  given  orally  by  the  teacher. 

§ 80,  the  definition  of  ratio,  should  be  stated  and  explained 
by  the  teacher.  It  may  be  explained  by  the  teacher  that  as 
a quotient  expresses  how  many  times  the  divisor  is  contained 
in  the  dividend,  so  the  ratio  of  two  numbers  means  how  many 
times  (either  integral  or  fractional)  the  first  number  contains 
the  second,  e.g.,  to  state  that  the  ratio  of  two  magnitudes  is 
6 3 (f  > or  2)  means  that  the  first  is  two  times  the  second, 

or  twice  as  large.  To  say  that  the  ratio  is  3 to  4 (or  |) 
means  that  the  first  is  | of  the  second,  or  f as  large. 

Problem  i may  be  written  out  at  home  after  one  or  two 
parts  are  done  in  class. 

Problems  2 to  8 can  be  done  orally  in  class  to  develop 
the  meaning  of  the  ratio  of  one  number  to  another  as  explained 
above. 

Lesson  2:  through  page  112 

Page  no.  Problems  i to  5,  may  be  used  as  a class  labora- 
tory exercise.  With  protractors  to  construct  the  angles 
have  all  pupils  draw  the  triangles  specified  in  the  problems. 


48 


First-Year  Mathematics  Manual  [pp.  iio- 


The  triangles  will  not  be  of  the  same  size  but  if  carefully 
drawn,  will  be  of  the  same  shape.  When  measured,  the  ratio 
of  any  two  corresponding  sides  of  two  triangles  having  equal 
angles  will  be  found  to  be  {approximately)  the  same  as  that 
of  two  other  corresponding  sides.  Have  the  pupils  under- 
stand that  such  lines  and  their  measurements  are  always 
approximate  and  get  them  to  draw  lines  and  measure  them 
with  as  little  error  as  possible.  § 82  states  that  such 
triangles  as  were  drawn  in  § 81  with  corresponding  angles 
equal  will  be  of  the  same  shape,  and  are  called  similar 
triangles. 

Problem  i may  be  done  on  the  board  by  the  teacher, 
changing  the  unit  from  inches  to  feet. 

Problems  2,  3,  and  4 may  be  assigned  as  home  work. 

§83  calls  attention  to  the  facts  which  are  formulated 

in  § 84. 

Problems  i to  12  show  some  applications  of  the  facts 
regarding  similar  triangles.  Problem  i jnay  be  worked  on 
the  board  by  the  teacher. 

If  ^ = 12,  and  a = 4,  the  ratio  of  corresponding  sides  of 
the  larger  triangle  to  the  smaller  is  3,  5 is  3X6,  or  12. 

Problem  2 may  be  made  a class  laboratory  exercise;  3, 
4,  and  5 may  be  assigned  as  home  work. 

Lesson  3:  page  113  to  § 85,  page  115 

The  teacher  may  ask  the  questions  in  Problem  6. 

To  prove  the  triangles  similar  it  must  be  assumed  that 
BA||KH  and  ZK=ZA,  and  ZH=ZB.  The  ratio  of 
corresponding  sides  is  3. 

* = 3Xi§,  or  5 

Let  the  teacher  conduct  the  class  through  Problems  7 
and  8.  Explain  the  meaning  of  Problem  9 and  assign  9 to 
12  as  home  work.  It  is  evident  that  in  similar  triangles  the 
shortest  side  of  one  triangle  corresponds  to  the  shortest  side 


-iig]  Problems  in  Proportion  and  Similarity 


49 


of  the  other,  and  so  on.  Assign  14  and  15  as  home  work. 
Problem  10  the  teacher  will  draw  on  board  and  develop. 
Impress  again  the  need  of  accuracy  in  drawing  measurements. 
In  Problem  15  have  all  the  parallels  on  the  triangle  at  the 
same  time.  Measure  and  find  ratio  of  parts  included 
between  the  same  parallels. 

Lesson  4:  § 85,  page  115,  to  § 91,  page  120 

The  teacher  will  lead  the  class  to  see  that  § 85  is  a formal 
statement  of  the  facts  developed  in  the  preceding  problems. 
Have  pupils  learn  these  two  theorems. 

Work  Problems  i and  2 with  the  class.  The  ratio 

or  f.  Thus,  A E,  or  x,  is  5. 

The  solutions  of  i and  2 depend  on  the  fact  that  the 
corresponding  angles  of  the  triangles  are  equal  (see  §81). 
Teacher  should  make  clear  the  meaning  of  § 86  and  have 
pupils  memorize  it  for  future  use.  Assign  Problem  i as 
home  work.  Teacher  should  develop  Problem  2 on  board, 
selecting  those  ratios  more  easy  to  use,  and  show  that  the 
result  as  stated  in  § 87  is  the  converse  of  § 85,  second  part. 

The  teacher  may  read  §§88  and  89  and  make  clear  the 
meaning  of  the  bearing  of  a line.  Have  pupils  do  Problem  i 
orally,  and  Problem  2 on  the  board.  Teacher  should  explain 
§90  and  pupils  do  Problem  i orally.  Teacher  may  do 
Problem  2 on  the  board,  assign  3,  4,  S,  and  6 for  home  work. 

Lesson  5:  § 91  /(?  § 94,  page  123 

Let  the  teacher  call  attention  to  the  transit.  Fig.  13 1. 
Degrees  are  marked  on  the  vertical  wheel,  for  getting  angle 
of  elevation  and  depression,  and  on  the  horizontal  wheel  for 
getting  bearings,  etc.  The  various  thumb  screws  are  to 
make  minute  changes  of  level  or  direction  of  the  telescope. 


5 o First-  Y ear  M athematics  M anual  [ pp . 120- 

The  angle  measurer  described  in  § 91  may  be  constructed  by 
the  teacher  and  shown  to  the  class,  the  pupils  to  be  en- 
couraged to  make  their  own,  and  use  them  in  original 
problems  like  those  on  pages  121  and  122.  The  two  small 
objects  in  the  corner  are  levels  to  determine  when  the  board 
is  horizontal. 

Assign  Problems  i and  2 for  home  work,  to  give  practice  in 
scale  drawing,  and  in  Problem  2 to  show  that  if  two  tri- 
angles have  two  sides  and  the  included  angle  in  one  triangle 
equal  to  those  of  the  other  the  remaining  side  in  the  one 
triangle  is  equal  to  that  in  the  other.  Work  Problem  2 with 
the  class  (they  on  paper)  (B  A = i99|  ft.). 

The  similarity  of  the  triangles  is  shown  by  § 87  and  § 86, 
supposing  that  the  triangle  in  the  drawing  to  be  placed  on  the 
surveyed  triangle,  or  use  second  paragraph  of  § 83. 

Each  pupil  may  be  assigned  one  problem  in  each  of 
§§91,92,93. 


Lesson  6 : to  Problem  6,  page  I2y 

The  teacher  should  draw  Fig.  139  on  the  board  and  show 

that  the  areas  are  and  ^h:  the  ratio  is  ^ or  J.  The 

ratio  of  the  bases  is  also  note  that  this  is  because  the  alti- 
tudes are  equal. 

Go  over  orally  with  the  class  Problems  2 to  7.  In 
Problem  8,  page  124,  have  the  class  tell  different  methods 
of  finding  x:  (i)  by  solving  the  equation  4^  = 40,  (2)  by 
reducing  the  ratio  to  have  denominator  8,  (3)  by  reducing 
each  ratio  to  denominator  4,  (ix)  = s,  x=io. 

Some  of  Problems  9 to  13  may  be  assigned  as  home  work. 
In  taking  up  § 95  call  attention  to  the  fact  that  equating 
the  ratios  in  Problems  8 to  13  formed  proportions.  The 
second  paragraph  of  § 95  may  be  stated  in  the  form:  lines  are 


-128  ] Problems  in  Proportion  and  Similarity  51 

proportional  if  the  numbers  expressing  their  lengths  form  a 
proportion. 

Problem  2.  (i)  is  shown  in  § 94,  i,  Fig.  139. 

(2)  May  be  shown  by  Fig.  139,  taking  EH  and  AD  as 
bases, 

(3)  Construct  III  with  a = a,  and  b — b. 

Then  ^ = blb',  and  ^ =ala'. 

Divide  first  equation  by  second,  member  by  member, 
l/ll  = ab'/a'b, 

(4)  Diagonals  drawn  on  figure  for  (3)  will  furnish  tri- 
angles that  will  prove  (4)  in  the  same  way  as  (3). 

If  we  assume  that  the  area  of  a rectangle  is  equal  to  the 
product  of  base  and  altitude,  (3)  may  be  demonstrated  thus: 
A=ab,  A'  = a'b',  Divide  first  equation  by  second  equa- 
tion Aj  A'  = abl  a'b' , 

Problem  3 may  well  be  explained  by  the  teacher  and  he 
should  call  attention  to  the  law  that  a product  is  divided 
by  any  number  when  one  of  its  factors  is  divided  by  that 
number.  In  getting  J of  the  area,  we  can  divide  the  4 by 
2,  and  the  15  by  3 and  get  integral  results.  Let  the  class 
for  home  work  solve  as  many  as  they  can  of  Problems  4-1 1. 

Lesson  7:  Problem  12,  page  127,  to  Problem  ii,  page  128 

Let  § 96,  Problems  i and  2,  be  gone  over  orally  with  the 
class,  and  the  teacher  test  the  class  on  them  the  next  day. 
Note  that  only  those  expressions  in  which  the  ratios  are 
equal  are  proportions.  Thus  in  Problem  2,  (3)  and  (5)  are 
not  proportions. 

Let  § 97  be  memorized  for  future  use. 

Let  Problem  i be  oral  work.  Problem  2 is  proved  by  the 
multiplication  axiom,  the  fundamental  law  for  the  reduction 
of  equations  containing  fractions.  Multiply  each  fraction 
by  bd. 


52 


First-Year  Mathematics  Manual  [pp,  128- 


Develop  Problem  3 and  emphasize  the  importance  of 
the  knowledge  of  ratio  in  the  solution  of  practical  problems. 

First  method:  As  the  parts  are  in  the  ratio  2 to  3,  20:  and 
2^x  will  represent  them,  giving  the  equation  3X+ 2^  = 85, 


x=i7,  2x  = 34,  3:^  = 5!. 

Second  method:  Let  x be  one  part,  then  S^—x  is  the 

X 

other.  As  they  are  in  the  ratio  2 to  3 then  = f . Multi- 


ply each  side  by  3(85— a;),  or  equate  the  product  of  the  ex- 
tremes and  the  product  of  the  means  (see  § 97),  and  get  by 
either  process  3:^=170—20;,  then  o;=34  and  85— o;  = 5i,  the 
required  parts.  Check:  ff  = 34+5^  =^5* 

Problem  4.  3o;+4o;+5x  = 84,  i2o;  = 84,  o;=7  whence3x  = 
21,  4X  = 28,  50;  = 35,  the  required  numbers.  Check:  = 

or  7 = 7 = 7,  and  21+28+35  = 84,  or  |i=f,  ||=|, 

Problem  5.  Let  x be  the  required  number.  Then 


Problem  6.  7— — =f,  o;  = 6,  S^=3o,  60; =36. 

60;— 12  J ^ 7 ^ 

Problem  7.  Let  x be  number  of  cm.  in  the  first  part, 

then  30—0;  is  the  number  in  the  second  part.  — 

30  3f 

or  or  f,  then  o;=2o,  30— o:=io. 

Second  method:  3fo;=7S.  6jo:=30,  o:=4|,  2\x=i2, 

3fo;=i8. 


Problem  8.  Let  0;  = number  of  inches  in  the  length  of 
AD.  Then  the  number  of  inches  in  D C = o:+2.  Whence 


o;+2=4. 

o;"t"2 

Problem  9.  Lay  off  A D = 2 cm.  and  D C = 4 cm.  With 
compasses  set  at  3 cm.  and  with  A as  center  strike  an  arc. 
With  C as  center  and  compasses  set  at  6 cm.  intersect  the 


-I2g]  Problems  in  Proportion  and  Similarity 


S3 


first  arc  as  at  B,  connect  B and  D and  with  protractor 
measure  angles  A B D and  D B C.  They  should  be  equal. 
This  illustrates  a law  of  geometry  that  a line  from  the  vertex 
of  an  angle  of  a triangle  bisects  the  angle  if  the  line  divides 
the  opposite  side  into  parts  proportional  to  the  sides  includ- 
ing the  angle. 

Problem  lo.  Let  x be  the  number  of  degrees  in  one  angle, 
then  90— X is  the  number  in  the  complementary  angle. 

Hence  — - — = then  x = 26,  go— ^ = 64.  Check:  ^ 

90— x+6  ^ ^ 64+6 

= f,  also  26+64  = 90. 

Assign  these  problems  as  home  work,  and  have  pupils 
put  them  on  the  board  next  day.  Require  the  check  as  an 
essential  part  of  the  work. 


Lesson  8:  Problem  ii,  page  128,  to  § 98,  page  i2g 


Give  the  class  drill  on  the  problems  of  § 97  in  forming 
the  proportions  or  equations  in  the  solution  of  the  equations, 
and  in  the  checking  of  the  results. 

Problem  ii.  Let  x and  180— x be  the  number  of  degrees 

in  the  angle  S.  Then  577^7—;:^  = i or  ^ 


120, 


180— x = 6o. 

2 • 120 


8(180— x) 


4(180— x) 


= 1;  :»:  = 


Check: 


8 


, =-7-  = !,  also  120+60  = 180. 
60  4^0 


Problem  12.  2x+3X+4X  = 36o,  x = 4o,  2X  = 8o,  3X  = 

120,  4x  = 160. 

Check:  or  40  = 40  = 40.  Also  80+120+ 

160  = 360. 


Problem  13.  x+2x+3x=i8o.  (The  sum  of  the  angles 

of  a triangle  is  i8o°.)  x = 3o.  The  3 angles  are  30°,  60°,  and 
90°. 

Problem  14.  2x+5x  = 90.  x=i2f,  2x  = 2sf,  5x  = 64|. 


54  First-Year  Mathematics  Manual  [pp.  i2g- 

Problem  15.  (5)  2o-\-x—x^  = ()—x—x^j  20:=— 14,  x — 

Check:  ^ = -~,  -3  = -3- 
4 3 

(10)  = 64X400.  Take  sq.  root  of  the  equation,  x = 
8X20  or  160. 

Check:  = 4^  — 4 0ji  ^ 400  = 400. 

x^ 

(11)  — =tV2=¥C  » 64x^  = 27.  Extract  cube  root,  4^  = 3, 
Check: 

Multiply  each  side  by  3,  = or  divide  numerator 

27/ 102 

and  denominator  of  first  fraction  by  3,  ^ =tV2~  ? TF2 

(12)  (:r+5)^  = 4Xi6.  Take  square  root,  a;+5  = 2X4, 
^ = 3* 

Check:  f = -VS  2 = 2. 


Lesson  9:  page  i2g  through  page  131 

§ 98  follows  from  § 97,  second  paragraph,  for  one  pair 
of  factors  of  the  equal  products  is  made  the  means,  and  the 
other  pair  the  extremes. 

Note  that  Problem  i,  equation  (i)  is  not  a proportion  for 
the  numbers  are  products.  (6)  to  (9)  are  not  proportions. 

The  teacher  may  work  with  the  class  Problems  i to  6, 
also  one  proportion  for  6,  and  assign  6 and  7 for  home  work. 
§ 99  is  a formal  statement  or  theorem  derived  from  the  pre- 
ceding work.  Give  class  practice  on  the  six  parts  of  Problem 
I,  and  assign  some  for  home  work.  Use  § 100  as  a short  class 
exercise,  and  assign  as  home  work. 


-132  ] Problems  in  Proportion  and  Similarity 


55 


In  §101  the  teacher  should  lead  the  pupils  to  see  that 
the  first  proportion  states  the  ratio  of  two  sides  of  one  tri- 
angle = ratio  of  the  two  corresponding  sides  of  the  larger 
triangle.  This  is  true  because  B E ||  C D,  and  the  triangles 
are  similar. 

In  proportion  2,  §101,  the  means  have  been  alternated, 
and  the  result  shows  that  the  ratio  of  two  corresponding 
sides  of  the  two  triangles  equals  the  ratio  of  two  other  corres- 
ponding sides. 

In  Problem  2,  note  that  the  polygons  are  equilateral. 
In  proportion  2,  by  alternating  the  means  the  corresponding 
sides  are  seen  to  be  proportional. 

Problem  3.  The  truth  of  proportion  i comes  from  the 
fact  that  opposite  sides  of  a parallelogram  are  equal.  Then 
take  proportion  i by  alternation.  Let  § 102  be  class  work 
by  teacher  and  pupils. 

Lesson  10:  Variation,  page  132,  to  Problem  5,  page  133 

Test  the  class  on  Lesson  9.  Teacher  will  draw  diagram 
(Fig.  151)  on  the  board  in  the  presence  of  the  class,  step  by 
step,  and  have  class  answer  the  questions.  D E is  a straight 
line.  The  angles  of  the  triangles  are  mutually  equal.  The 
ratios  are  each  2 to  i. 

Problem  2.  The  triangles  are  similar  because  mutually 
equiangular,  and  the  corresponding  sides  are  proportional. 
(The  ratio  of  each  vertical  side  to  the  hypotenuse  is  Vj 
found  by  the  law  of  the  right  triangle.) 

The  ratio  of  the  distance  line  to  the  time  line  is  2 to 
doubled,  trebled,  etc.  The  distance  line  varies  (changes)  as 
the  time  line  varies. 

Teacher  develop  with  the  class  Problem  3.  d = 2t,  d = 
2,  4,  6,  8 ...  . 20.  The  ratio  of  J to  ^ is  2.  (Divide 
each  side  of  equation  d = 2t  by  t,) 


S6 


First-Year  Mathematics  Manual  [pp,  ij2- 


Assign  Problem  4 as  home  work. 

The  cost  doubles,  trebles,  etc.,  as  the  weight  doubles, 
trebles,  etc.  The  cost  varies  as  the  weight.  The  ratio  of 
cost  to  weight  is  3!^. 

Assign  Problems  5,  6 for  home  work. 

§ 102,  Problem  i:  Teacher  develop  Problems  i and  2. 
Let  horizontal  side  of  small  square  be  the  base,  and  vertical 
side  the  area.  The  area  is  then  3 times  the  base  in  each 
triangle.  A varies  as  the  base. 

Problem  2.  A=~  , A=bX-  , and  if  h (and  therefore  - 
22  2 

A h 

is  constant,  l^he  ratio  of  A to  b is  constant,  and 

A varies  as  b. 

Assign  Problem  3 for  home  work;  also  Problem  4.  State 
to  class  what  Problem  4 calls  for,  but  do  not  tell  what 
the  ratio  is.  (The  ratio  is  approximately.  See  Problem 
6,  page  134.) 


Lesson  ii:  page  133  through  page  133 


Read  Problem  5,  and  get  class  to  make  the  equation. 


— = ^,  ox  A = kb ^ that  is,  the  area = base  times  a constant 
b 

number.  Then  ^^-  = k = g or  the  area  is  9 times  the  base. 
When  the  area  is  54,  $4  = gXb,  therefore  the  base  is  6. 
Assign  Problems  6,  7 ....  1 1 for  home  work. 


Explain  § 103. 


This  means  that  — r-  = ^ a constant,  or 
i/a  ' 


simplified  ab  = k. 


Assign  Problem  2 for  home  work.  Develop  Problem  3. 
Explain  the  statement  about  compression  of  gas.  Make 
clear  that  it  is  not  simply  illuminating  gas  that  is  meant  but 
the  state  of  matter  represented  by  air,  hydrogen,  steam,  etc. 


-ij6  ] Problems  in  Proportion  and  Similarity 


57 


Take  ice,  water,  and  vapor  of  water  (or  steam)  to  illustrate 
the  three  states  of  matter — solid,  liquid,  and  gaseous.  The 
equation  is  vp  = k,  hence  4X 3 = or  yfe  = 1 2.  Then  when  the 
pressure  is  6 lb.,  v • 6 = 12,  or  the  volume  is  2.  Assign  the 
rest  of  these  problems  for  home  work. 

In  Problem  8 explain  that  the  time  a pendulum  vibrates 
once  depends  on  its  length.  Make  an  impromptu  pendulum 
of  string  and  a small  weight  (bunch  of  keys),  and  have  class 
observe  the  difference  in  time  due  to  different  lengths  of 
string.  This  relation  of  time  and  length  is  stated  in  the 

problem,  and  is  expressed  exactly  in  the  equation  = k,  and 

y I' 

t = k • 1/7  when  the  length  is  expressed  in  inches  and  time 
in  seconds. 

According  to  Problem  8,  i =^1/39. 2 or  i=^X67  (ap- 
proximately), ^ = For  ^ pendulum  vibrating  in  2 seconds 
2 = ^1/;^  6^2  Qj.  pendulum  is  153.76 

inches  long. 


Lesson  12:  page  136,  Summary 

Review  the  pupils  on  the  definitions  in  black-faced  type, 
and  on  the  theorems  in  italics.  Also  give  problems  to  test 
their  grasp  of  the  subjects  and  of  the  processes  treated  in  the 
chapter. 


\pp.  ijy- 


CHAPTER  VII 

PROBLEMS  ON  PARALLEL  LINES;  GEOMETRIC 
CONSTRUCTIONS 

Lesson  i : through  page  ijg 

The  teacher  should  introduce  the  work  to  the  class,  and 
have  the  class  draw  the  lines  and  make  the  measurements  of 
Problems  i to  4.  Have  the  class  regard  § 105  as  a provi- 
sional definition  of  parallel  lines. 

Teacher  show  how  to  solve  Problems  i and  2.  Assign 
Problems  3 and  4 for  home  work.  Require  pupils  to  fasten 
to  their  work  the  paper  angles  used.  Ask  pupils  to  note  and 
remember  theorems  (a),  (6),  and  (c)  of  § 106. 

Teacher  will  draw  Fig.  157  on  the  board,  and  have  the 
pupils  answer  questions  in  Problems  2 to  6 as  in  i.  Assign 
Problems  7 to  13  as  home  work. 

Lesson  2:  page  140  to  § 109,  page  142 

§ 107.  Draw  Fig.  159  on  the  board,  and  drill  pupils  on 
naming  the  various  angle  pairs.  Drill  on  Problem  i. 

Draw  Fig.  160  on  board,  and  have  pupils  give  proofs  of 
Problems  2 and  3. 

Assign  Problems  4,  5,  6,  and  all  of  § 108  as  home  work. 

Problem  i.  Page  141,  yxD  is  supplement  of  x,  y=yxD 
(§  106,  Problem  8),  is  supplement  of  x.  Similarly  for  y 
and  2. 

Problem  2.  2 is  supplement  of  y.  x is  supplement  of 

y,  z = x. 

Problem  3.  a:+y=i8o°.  ■z£;-l-2  = i8o°.  x-\-y-]rW-\-z  = 

360". 


58 


Geometric  Constructions 


59 


-143] 


Lesson  3:  § 109,  page  142,  to  middle  of  page  14J 

As  a general  class  exercise  Theorems  I and  II  should  be 
shown  to  follow  from  the  previous  problems  on  parallel 
lines,  and  remembered  for  future  use. 

§110,  Problems  i and  2,  can  be  shown  to  follow  from 
previous  problems  on  parallel  lines.  Assign  Problems  3 to 
9 as  home  work,  say  certain  different  groups  of  three  or  four 
problems  to  each  pupil. 

Problem  5 follows  from  § 106  (b). 

Problem  7.  /+y+ze;=i8o°  (a  straight  angle)  but  t = 

X and  w=z  (§  106,  Problem  8).  Substituting  x for  t and  z 
for  w,  then  x+y+z=  180°. 

Problem  8.  In  the  diagram  w-\-y=  2KA 

U+X=2RA 
t+z  = 2RA 

Adding,  exterior  Z^+interior  Z5  = 6RA 
Sum  of  interior  Z5  = 2RA 
Subtracting,  sum  of  exterior  Z5  = 4RA 

Problem  9.  a+&+c  = 2RA  (see  Problem  7) 

c+z  = 2RA  (straight  angle) 
a+b+c  = c+x  (equality  axiom) 

Subtract  c,  a-]rb  = x. 

Lesson  4:  Algebraic  exercises,  page  14 j to  § m,  page  145 

This  lesson  should  serve  to  fix  in  mind  the  laws  of  angles 
formed  by  parallel  lines  and  a transversal,  and  give  practice 
in  the  solution  of  equations  of  the  first  degree. 

Work  Problem  i with  the  class  and  assign  Problems  2 to 
II  for  home  work,  say  a certain  (different)  five  or  six  to 
each  pupil. 

Problem  i.  2(4^— 3)  = 79+3:1^  (corresponding  angles  of 


6o  First-Year  Mathematics  Manual  [pp.  14J- 

^lines).  Then  8x— 6 = 79+3x;  Check:  79+3a;=79 

+51  = 130,  8x— 6=  136  — 6 = 130.  The  adjacent  angle  = 5o°. 

Problem  3.  3X— 5 = s(x— 7)  (alternate  interior/ 5 of 
y/  lines). 

Lesson  5 : page  145  through  Problem  15,  page  146 

§ III.  Problems  on  Constructions 

Show  on  board  how  to  do  Problem  i.  Then  have  class 
practice  at  their  desks  with  compasses  and  straight-edge 
till  they  can  draw  at  a given  point  on  a straight  line  (drawn 
in  any  direction  on  the  paper)  a perpendicular  to  the  line. 

Assign  Problems  2 to  7 for  home  work.  Direct  the  pupils 
to  make  the  construction  lines  plain  and  not  to  erase  them. 

Teacher  show  how  to  do  Problem  8 and  make  clear  that 
either  point  G or  point  F may  be  used,  but  only  one  is  needed. 

Problem  9.  A B must  be  prolonged. 

Give  Problems  10  to  13  as  home  work. 

In  Problem  10  explain  that  the  word  altitude  used  here 
means  the  altitude  line  drawn  from  the  vertex  of  the  angle. 

Lesson  6:  to  Problem  24,  page  148 

Teacher  show  class  how  to  do  Problem  15.  Have  class 
practice  at  their  desks  bisecting  given  lines  drawn  in  any 
direction  on  the  paper. 

Assign  Problems  16,  17,  and  18  (or  two  of  them)  for  home 
work.  Teacher  show  Problem  19  and  have  class  practice. 

Assign  Problems  20  to  23  for  home  work. 

Problem  20.  The  bisectors  should  meet  at  a common 
point. 

Problem  21.  The  sum  of  the  halves  of  the  two  parts  of 
a straight  angle  equals  a right  angle. 

Problem  22.  The  halves  of  equals  are  equal,  i.e.,  i = 
but  if  corresponding  angles  are  equal  the  lines  are  parallel. 


Geometric  Constructions 


6i 


-151] 

Problem  23.  a and  b are  halves  of  supplementary  angles 
or  I of  180°,  or  90°,  as  the  sum  of  x,  a,  and  b is  180°,  then  x = 
90°  and  the  lines  are  perpendicular. 


Lesson  7:  to  Algebraic  Exercises,  page  150 

Teacher  show  Problem  24,  and  have  class  practice. 

Problems  25  to  30  give  practice  in  drawing  an  angle 
equal  to  a given  angle,  and  some  or  all  may  be  assigned  for 
home  work. 

Problem  26.  Draw  three  angles  at  the  same  vertex  as 
adjacent  angles.  The  sum  is  180°,  and  the  exterior  sides 
should  form  a straight  angle,  or  a straight  line. 

Problem  27.  Let  A B C be  any  given  triangle.  Draw 
indefinite  line  XY.  Lay  off  a segment  A'C'  = AC.  Con- 
struct at  A',  XA'D=ZC  and  lay  off  A'D  = CB.  At  D 
draw  an  angle  A'DE=ZC,  and  at  C'  an  ZA'C'E  = C. 
A'C'E  D is  the  required  parallelogram,  as  the  opposite  pairs 
of  sides  are  parallel. 


Lesson  8:  to  Summary,  page  151 

The  nine  problems  give  practice  in  the  solution  of  equa- 
tions of  the  first  degree,  formed  by  applying  the  laws  of  geo- 
metrical figures.  Explain  one  or  two  if  found  necessary 
and  assign  all  as  home  work. 

Problem  3.  2y+22  = 3y— 2,  and  2y+22+2o  = 180°. 

Solving,  2 = 20  and  y = 6o.  The  obtuse  angle  = 160°. 

Problem  4.  3ji;+5x=i8o,  x = 22\.  8y+2y  = i8o,  y=i8. 

The  :r-angles  are  67!°  and  112^°. 

The  y-angles  are  144  and  36. 

Problem  5.  The  angle  not  lettered  = angle  x.  (Alter- 
nate interior  angles  of  parallel  lines.)  ^ = 45  and  y=i3S. 


62 


First-Year  Mathematics  Manual  [pp.  iji- 


CC  0C~\~  1 2 

Problem  6.  From  similar  triangles  — = , x=i6.8, 

14  24  ^ 

y 'V+io 
— , 3'  = i4- 

14  24  ’ ^ 

ai 

Problem  7.  — , — = , i2:r— i = 10X+4  , x = 2h, 

5X+2  I2X— I 


3y±3^sy±9 


\^y+^=sy+9^  3^=3^ 


Problem  8. 


^3  ^ 9+a 

iS+6 


45+3^  = 45+5^)  ^ = 


c=  12 


+ i|a. 

The  area  of  the  trapezoid  is  found  by  subtracting  the 
small  triangle  from  the  whole  triangle  of  the  figure  194. 

The  whole  triangle  is  Ji2+ia){g+a)  ^ 

(6+f  a)  (9+ff)  = 54+ 1 2aH-| 

The  area  of  the  small  triangle  is  54,  area  of  trapezoid  is 
i2a+fa. 


Lesson  9 : Summary 
Make  this  a review  of  the  chapter. 

Test  pupils  on  all  definitions,  theorems,  and  constructions, 
and  on  problems  depending  on  them,  by  oral,  blackboard, 
and  written  work. 


CHAPTER  VIII 


THE  FUNDAMENTAL  OPERATIONS  APPLIED  TO 
INTEGRAL  ALGEBRAIC  EXPRESSIONS 

Notice  the  title  of  this  chapter.  By  the  “fundamental 
operations’’  is  meant  algebraic  addition,  subtraction,  multi- 
plication, and  division.  The  teacher  who  believes  the  child 
is  to  get  a knowledge  of  these  operations  by  performing 
meaningless  mechanical  exercises  in  adding,  subtracting, 
etc.,  at  a stage  so  early  that  he  cannot  yet  have  any  real  con- 
cept of  algebraic  numbers,  much  less  any  real  feeling  as  to 
why  such  numbers  should  be  added,  subtracted,  etc.,  will 
of  course  feel  this  chapter  to  be  placed  rather  late  in  the 
year.  The  stereotyped  plan  is  against  it,  and  most  teachers 
have  had  this  training  only  under  this  plan.  It  is  reflecting 
on  no  one  and  is  perhaps  even  asserting  nothing  new  to  state 
that  the  promptings  of  early  training  have  more  to  do  with 
the  teachers’  notion  of  what  algebra  is  and  of  why  and  how 
it  should  be  taught  than  do  all  other  influences  combined. 
A few  words  by  way  of  justifying  the  placing  of  this  chapter 
here  and  of  showing  the  reason  for  the  plan  of  treatment  may 
be  helpful.  This  development  of  the  mathematical  intuitions 
and  powers  of  first-year  high-school  pupils  is  based  frankly 
upon  the  doctrines  that: 

I.  The  work  should  appeal  to  the  interest  of  the  learner 
and  should  court  the  sanction  of  his  judgment  that  it  is  worth 
while  to  master  it. 

II.  This  can  be  done  only  by  appealing  much  more  to  his 
understanding  than  to  his  memory  and  thereby  securing  a 
real  insight  into  the  meanings  and  reasons  for  what  he  is 
called  upon  to  do. 


63 


64 


First-  Y ear  M athematics  M anual  [ pp . 15 j- 


III.  The  more  practical  and  untechnical  parts  of  the 
beginnings  of  algebra  and  geometry  furnish  the  best  approach 
to  mathematical  science,  because  they  create  the  most 
natural  and  the  friendliest  attitude  of  mind  toward  mathe- 
matical study. 

The  pages  that  precede  this  chapter  have  reconnoitered 
the  fields  of  generalized  arithmetical  number  and  of  rational 
mensuration,  have  extended  by  pictured  representation  and 
common-sense  illustration  the  notion  of  numbers  to  include 
the  negative,  have  enriched  the  extended  notion  by  arith- 
metical and  geometrical  uses  in  problems  treated  more  or 
less  informally,  and  have  carried  the  evolution  of  ideas  to  the 
point  where  technical  algebraic  operations  are  needed  for 
further  advance.  Up  to  this  point  the  learner’s  arithmetical 
knowledge  and  skill  has  been  the  main  reliance;  he  is  ready 
now  to  go  beyond  with  understanding  and  profit. 

This  chapter  should  be  divided  into  nine  lessons  about  as 
follows: 

Lesson  i:  from  § 112,  page  153,  to  Exercise  XVI,  page 
156. 

Lesson  2:  from  § 115,  page  156,  to  § 119,  page  159. 

Lesson  3:  from  § 119,  page  159,  to  § 121,  page  162. 

Lesson  4:  from  § 121,  page  162,  to  § 123,  page  166. 

Lesson  5:  from  § 123,  page  166,  to  § 125,  page  169. 

Lesson  6:  from  § 125,  page  169,  to  § 127,  page  172. 

Lesson  7:  from  § 127,  page  172,  to  Exercise  XXXI, 
page  176.  Solve  one-half  of  Exercise  XXXI  in  class  and 
one-half  as  home  work. 

Lesson  8:  from  Exercise  XXXI,  page  176,  to  Exercise 
X,  page  179. 

Lesson  9:  from  Exercise  X,  page  179,  to  end  of  chapter. 

Lesson  10:  Review  of  addition,  subtraction,  multipli- 
cation, and  division,  each  pupil  solving  in  class  or  at  home 


The  Fundamental  Operations 


6S 


-154] 


and  bringing  work  to  class  at  least  one  moderately  difficult 
example  in  every  one  of  the  four  fundamental  operations. 
It  would  be  well  to  make  Lesson  10  a written  recitation. 

If  time  is  short  Lesson  10  may  be  omitted. 

In  every  case  it  will  be  better  to  omit  certain  indicated 
exercises  within  the  assignment  than  to  shorten  the  assign- 
ment. Pupils  will  learn  more  to  go  over  the  work  at  about 
the  rate  of  progress  indicated  by  the  foregoing  assignments 
than  to  go  more  slowly.  It  is  serious  educational  waste,  not 
to  mention  loss  of  interest,  to  mistake  slow  progress  for  thor- 
oughness. Overmature  ideals  of  thoroughness  have  no 
place  in  a first-year  high-school  class  in  mathematics.  If 
teachers  of  mathematics  could  bring  themselves  to  believe 
and  to  practice  this,  secondary  mathematical  teaching 
would  at  once  undergo  marked  improvement.  The  exercise 
with  which  §112  opens  shows  under  Solutions  I and  II  the 
advantage  of  grouping  addends  in  algebraic  fashion  into 
sums.  Solution  II  surely  impresses  the  pupil  with  the  great 
gain  of  adding  first  the  coefficients  and  then  multiplying  by 
the  common  factor.  Have  some  pupil  read  aloud  in  class 
and  actually  verify  all  the  multiplications  and  additions  of 
Solutions  I and  II. 

Exercise  I,  at  the  bottom  of  page  153,  exhibits  a form 
of  the  problem  in  which  the  solution  comes  out  in  perfect 
algebraic  form  as  a polynomial,  all  of  whose  terms  are  then 
combined  into  a single  term,  440:^,  which  the  pupil  will  feel 
to  be  completely  known,  so  soon  as  it  is  known  what  number 
X stands  for,  and  also  the  pupil  will  hardly  miss  the  idea  that 
this  solution  is  applicable  to  any  such  problem,  no  matter 
what  the  particular  price  of  the  ticket.  He  can  sense  the 
interpretation  of  440X  as  the  value  of  the  440  tickets  each 
of  the  value  x cents.  That  is,  he  senses  440X  as  a number, 
rather  than  as  a mere  symbol,  or  as  a mysterious  ‘‘quantity.’’ 


66 


First-Year  Mathematics  Manual  [pp,  15^- 


Take  up  Problems  2,3,  and  4 orally  with  the  class.  Problem 
2,  page  154,  puts  conditions  in  a form  to  impress  the  pupil 
with  the  naturalness  of  associating  the  amounts  into  separate 
terms  and  then  adding  the  terms.  One  factor,  the  23.8, 
pervades  the  entire  problem. 

50  • 23.8  cents  + 17  • 23.8  cents+700  • 23.8  centsH- 
6150*23.8  cents,  or  (50+17+700+6150)  • 23.8  cents  = 
6917  *23.8  cents  = $1,646. 246. 

Problem  3 again  clothes  a problem  in  algebraic  garb. 

Solution:  65+8.s+4.y+65+io5  = (6+8+4+6+10)5“  = 345. 
Answer  345  ft. 

Problem  4.  6y+8y+ioy+i2y+i4y  = 503;.  Answer:  503; 
yards. 

§ 1 13.  Cite  from  problems  2-4,  or  have  pupils  cite, 
illustrations  of  the  meaning  of  “similar  with  respect  to  a 
factor.’’ 

Exercise  i.  The  parts  of  this  exercise  should  be  done 
orally  to  exercise  the  pupil  at  once  in  the  application  of  the 
definition  just  given. 

Answers:  (i)  Terms  are  similar  with  respect  to  x;  coef- 
ficients are  4,  —7,  20,  and  —35. 

(2)  Terms  similar  with  respect  to  x,  coefficients  a, 
— 25,  —6,  and  46. 

(3)  Terms  similar  with  respect  to  x,  coefficients  a,  —6, 
— c,  and  d. 

(4)  Terms  similar  with  respect  to  a,  to  x,  or  to  xa. 
Coefficients  i.  As  to  factor  a,  coefficients  2X,  3X,  — 7x,  — 5x. 

2.  As  to  factor  x,  coefficients  2a,  3a,  —7a,  —5a. 

3.  As  to  factor  xa,  coefficients  2,  3,—  7,  and  —5. 

(5)  Terms  similar  with  respect  to  coefficients  —3/^, 
6t,  — 8r,  and  125. 


The  Fundamental  Operations 


67 


-155] 

(6)  Terms  similar  with  respect  to  Xy  or  2,  or  xz] 
CoefiScients  i.  Common  factor  x,  coefficients  4^2,  --7C2, 

— 5J2,  and  gez, 

2.  Common  factor  2,  coefficients  /[ax,  —^cXy 

— ^dXy  and  gex, 

3.  Common  factor  xZy  coefficients  4a,  —7c, 

— SJ,  and  ge, 

(7)  Terms  similar  with  respect  to  m,  coefficients  aby 
—pq,  —xy,  and  —dc. 

(8)  Terms  similar  with  respect  to  aby  coefficients  3a, 
— ^\aby  and  —1.5^^. 

(9)  Terms  similar  with  respect  to  r , coefficients  aXy 
— 7.5^,  and  —iqbx, 

(10)  Terms  similar  with  respect  to  aPy  coefficients  —4/, 
—by  3X,  and  7.23;. 

Problem  2,  page  155,  Rule:  Add  the  coefficients  and  write 
the  common  factor  after  the  sum. 


Exercise  XV  (May  be  given  as  home  work) 


Polynomial  Simplified  Form 

1.  15a— 7a+i8a=  26a 

2.  — i8x^— i2x^+iS^^— — i8x^ 

3.  2\ab+2^\ab—^\ab  — ^lab=  — 4i^& 

4.  2Tabc—^^abc-\-ioabc  — 2abc=  o 

5.  s(a+b)-4(a+b)  + i2(a+b)=  ii(a+b) 

6.  — 8(x^+/)  — 24(x^+/)  + i7(x^+/)  = — isC^'^+y'*) 


7-  -3Up^-f)+5§(P^-f) -4Mp^-q")  = 

-2\^{pr-q^) 

8.  i8(w/>-35)"-is(m/>~35)"-37(w/>-35)^+i4(m/?~ 
35)2=  —2o{mp—2)Sy 

9.  a(x+y+2)— &(x+y+2)— c(x+3;+2)+f/(:r+y+2)  = 

((Z — b — c-\-d)  (x-|-y-t-2) 

10.  j^x^  — ffix^-{-Mx^  — ^cx^—{i^a—ffi-yM  — ^c)x^, 


68 


First-Year  Mathematics  Manual  [pp,  155- 


§ 1 14.  The  exercise  with  which  the  section  opens  is  to 
illustrate  addition  when  the  terms  are  dissimilar.  This  is 
to  enable  the  pupils  to  feel  the  reason  for  holding  the  separate 
terms  apart  as  in  algebraic  polynomials.  Do  I as  a general 
class  exercise  and  assign  II,  III,  and  2 for  home  work. 

Problem  I.  1.  (+3)(— i7)+(-5)(-i4)  + (-8)(-s) 

+ (+i2)(+i9)  = — 51+70+40+228  = 
+287 

II.  (+5)  (— a)+(+i2)  {—b)  +(+14)  (+6) 
+(— 6)  (+a)  = —5a— 126+146  — 6a  = 

— Iia+26 

III.  (-8)(-x)  + (+io)(-y)  + (-is)(-2) 

+ (+9)  {-\-w)  = ^x—ioy+i$z-\-gw 

Problem  2.  Answer,  8(a+6+c+t/)  steps. 

Exercise  XVI 

Do  one  or  two  with  the  class  and  assign  certain  others 
for  home  work. 

1.  ^sH—i2x^y-{‘T^^y—?)SH  = 2sH^^x^y 

2 . — 2 7 f + 1 8^c(/+ 1 5 + 1 Aicd  =—12  ^^ab + 3 2\^cd 

3.  ^ax—:^x-\-oo  = ^ax—2x 

4.  9a^6^--3cy +4a^6^— 4(;^y  — 3a^6^  = — 

5.  — 8w/>^+5a^ji£;— 3a^x— 4W^^+2a^:;t;=  — 

4a^x 

6.  — 4-^^ — 

7 . ar^p — r^p — br^p + pr^ — cpr^  = {a—b—c)  pr^ 

8.  27(a+6)'*+4c+isa— 12^;— i5(a+6)''  — 8a=i2(a+6)'' 

— 8c“|"7^ 

9.  a{a—b)-\rb{a—b)—c{a—b)  = {a-]rb—c){a—b) 

10.  a^{a+b)  — 2ab{a-Yb)  +6^(a+6)  = {a^—2ab-\-b'^) {a+b) 

11.  — 6j(a+6)+|(a^+6'*)— (^+^)+4§(^+ 
b)+(>h{a^+b^)  = (>l{a^+b^)  + l{a+b) 


-157]  The  Fundamental  Operations  69 

12.  r^{r+p)-y^{r+p)+y{r+p)  + {r+p)  = {r^-y^+ 
y+i){r+p). 

The  pupil  has  now  learned  through  use  and  application 
how  to  add  monomial  terms,  both  similar  and  dissimilar, 
and  how  to  simplify  them.  He  is  ready  to  apply  this  knowl- 
edge to  polynomials. 

§ 1 16.  This  section  begins  again  with  a simple  concrete 
problem  in  which  the  pupil  can  feel  the  real  meaning  of  the 
individual  terms  and  of  the  polynomials  as  wholes,  as  well 
as  the  sum  of  all  the  added  polynomials.  Do  not  think  that 
he  understands  merely  because  he  can  combine  the  coeffi- 
cients correctly.  Have  him  tell  the  meaning  of  everything 
he  does  in  this  problem.  Teachers  too  readily  mistake 
mechanical  imitativeness  for  insight. 

See  that  every  step  of  the  solution  of  i is  understood  and 
that  the  form  (+27^^^— iS:ry+i8y'^)  + (~i2X^+30Ji;:y— 3y^) 
is  conceived  as  the  sum;  but  that  isx^+i5xy+i5y^  is  the 
simplified  sum.  It  may  be  well  also  to  have  him  recognize 
2,s  another  form  of  the  simplified  sum. 


Exercise  XVH 

Have  pupils  do  all  they  can  orally,  writing  results  only. 
Answers: 

1.  Ta—jb-\-7c 

2.  — a+3c— 26+i3(/ 

3.  a'^+26'^— 3^^ 

4.  1 5xy— 12x^+293;^ 

5.  —14^+136  — 20c 

6.  —k-\r2l-\-4m 

7. 

8.  i^x^-2xy-z\y^ 

9.  4(a^+65)  — 2(a^+6*) 


70 


First-Year  Mathematics  Manual  [pp,  757- 


10.  ^{a-\-b-\-c)-{-T2i^  b-\-c)  — xo (”^“1"^ 
+c)  =^a+^b+2^QC 

11.  4p^+sP^+4p-4 

12. 

13.  —2ga^b-{-g2a^c-{-97c^b  — 2 2b^c 

14.  ga^{a+b)  — $a'^b{a^+b'^)  —4ab^{a^-\-b^) 

15*  il'^~\~()~Y(l — u^-\~'2‘tv. 

§ 1 17.  It  will  be  well  here  to  review  rapidly  and  orally 
the  exercises  on  page  76  to  remind  the  pupil  that  subtraction 
may  always  be  readily  changed  into  addition  by  changing 
the  signs  in  the  terms  of  the  subtrahend  from  + to  — and 
from  — to  +,  and  adding  the  changed  subtrahend  to  the 
minuend.  A pupil  will  repeat  this  as  a rule  long  before  he 
grasps  its  significance  to  consist  in  the  fact  that  after  reversing 
the  sign  of  the  terms  of  the  subtrahend,  the  laws  of  addition 
then  apply.  Have  the  pupil  here  see  again  that  by  reversing 
the  signs  in  the  subtrahend  and  adding  the  changed  subtra- 
hend to  the  minuend  he  does  obtain  a number  which,  added  to 
the  given  subtrahend,  gives  the  minuend.  The  pupil  has 
difficulty  in  freeing  himself  from  the  arithmetical  notions  of 
take  away  ’’  and  “less  than ’’  as  meaning  subtraction.  Take 
a little  care  here  to  insist  that  algebraic  subtraction  means 
finding  a number  (called  the  difference)  which,  added  to  the 
subtrahend,  gives  the  minuend.  This  gives  what  is  called  the 
difference  between  them.  Exercises  1-8,  rightly  solved, 
enable  the  pupil  to  sense  this  definition  a little.  Notice  the 
black-face  A and  S of  the  example,  meaning  add,  and  subtract. 

Answers: 

1.  32a;  4.  -2s{p+q)  7.  -2j^^{x-y) 

2.  —iia  5.  gm^px  8.  —iSm^{a  — 2b^). 

3.  siab  6.  ()x{i  + ^a^y) 

§ 1 18.  Read  carefully  with  the  class  the  matter  of  § ii8 
at  the  bottom  of  page  158.  Have  the  pupil  think  that  while 


The  Fundamental  Operations 


71 


~i6o] 


(+S^)“(~7^)  is  a difference,  the  form,  +i2X,  is  the  simpli- 
fied form  of  the  difference.  All  the  exercises  of  page  159 
are  differences  as  they  stand;  but  the  work  for  the  pupil  is 
to  simplify  them. 


Exercise  XVIII 


Answers: 

1.  —^ab 

2.  —2ia^rl^ 

3.  Sip^p 

4.  loos^gh 

5.  —i.ihk^v 

6.  12.  gp^q^s^ 

7.  —i2{a-{-b) 


8.  6(/^-/*3) 

9.  {a  — 6){m^+g) 

10.  {—x—^){v^—s'^) 

11.  {a—b){x-\-y) 

12.  ( — 2|a— — ay) 
13*  (a+ft+c) 

14.  (4.S5"-3.4/")(z)"-/^"). 


Notice  the  capital  S and  A standing  at  the  ends  of  the 
lines  under  the  formal  arrangements  at  the  bottom  of  page 
159.  They  stand  for  the  words  ‘^Subtraction’’  and  “Addi- 
tion” respectively. 

Page  160,  Answers:  i.  (i)  ^ab]  (2)  2a^6^+2a4-|-8a6^; 

(3)  x^-^6x^y-\-4y^ 

2.  (i)  — 4a6;  (2)  ^2a^b^  — 2a^  — Sab^]  (3)  — — 6x^y 

-4y3 

3.  2c^  — a^c-\-2d^ 

4.  x4-|-y^—4x^y— 4xy3+6x*y*. 


The  idea  of  Exercise  5 is  to  lead  the  pupil  to  use  his  head 
when  it  is  easier  and  more  expedient  than  to  use  the  pencil. 
Pupils  are  in  danger  of  forming  the  habit  of  wasting  time 
“in  mere  puttering”  when  they  are  allowed  or  encouraged  to 
use  pencil  and  paper  for  mere  trivial  steps  in  transformations. 
They  should  be  led  to  form  the  habit  of  looking  as  deeply  as 
possible  into  a problem  before  beginning  it,  and  then  doing 
as  much  as  they  can  with  reasonable  effort  mentally,  merely 
setting  down  results  of  the  mental  steps.  Pupils  of  ordinary 


72 


First-Year  Mathematics  Manual  [pp.  i6o- 


ability  will  easily  omit  the  intermediate  step  of  the  last  part 
of  the  solution  of  Exercise  5,  setting  down  —i^a^-\-T-A(Fb  — 
2a6^— 216^  at  once.  Continual  attention  to  this  practice 
will  enhance  both  the  interest  and  the  insight  in  the  work. 
This  matter  should  not  be  overdone,  as  of  course  it  may  be. 


Exercise  XIX 

Do  one  or  more  with  the  class  and  assign  others  for  home 
work. 

Answers: 

1.  6xy — 6y^ 

2.  — a^x+i26xy+3o&^y— 3oaxy 

3.  m^pq-\-'^^P'^'^^(f 

4.  2^^abc — 1 1 ^a^b  — i — Ic^a 

5 • 3 “ I — 2 ox^y^ 

6.  loy^— 3:^3— 7:r^y— 

7.  2\P — — S^l^m 

8.  3.4 

9.  42:^3  — 6:r^y — 8xy^+ 1 2y^ 

10.  ii.6t^-ishp-3y^f-s¥T 

11.  4og%—s6g^—Sgh^  — iSg^h^  — 23h^ 

12.  7 . 6.49^m^/+3 . 6kmH^—6kml^ 

13.  (3a+26-4):r3-(45+3/)y23+S23 

14.  {3m+4.n)uv  — {3m+3)v^+{A^—9^)^^- 

§ 120.  Problem  i.  A — sign  before  a polynomial  in 
parenthesis,  ( ),  requires  the  sign  of  every  term  of  the  poly- 
nomial to  be  reversed,  if  the  parenthesis  is  dropped. 

2.  If  a + sign  precede  a polynomial  in  parenthesis,  the 
parenthesis  with  its  + sign  may  be  simply  omitted,  no 
changes  being  made  in  the  signs  of  the  terms  of  the  poly- 
nomial. 


-i62\  The  Fundamental  Operations  73 

Exercise  XX 

Do  one  or  more  with  the  class  and  assign  others  for  home 
work. 

Answers: 

1.  = 

2.  8|-4^3+3^3-5§=3-^3 

3.  166^+42^^+36^+2  — 50^^— 3 = iie^  — I 

4.  2/-6/+3g-4/+2g-4/=5g-i2/ 

5.  = 

6.  gx-sy-\-(>y-{■^z-\-^y—4z  = gx+^y->r2,^ 

7.  — 3a''+4a3— 3a3-f-5a^_4a2_3a= 

8.  7 . S/>3 — 3 . 4/'’+4  • 2/>^+ 1 . 6/»"+3  • 4 • S/*  = 8 • 3/’^  + 

5/>^-4.5/) 

9.  3^^  — 2/>^ — ^^+/>^+r — /^^+r+^3+r  = 3^^  — 2/>^+3r 

10.  1055^— 145/— 3^^+45^—  2P  — 55^+25/  = 100^^  — 85/ — 5/^ 

11.  — 3r?+3r^— 3/^— 4r^+5r^— 2?+2r^+2/^  = 2r/+r^— 3? 

12.  — Sl^^+3  • 4^/^ — 2/^^ — 5 . \kh  +3^^  — 2 . 4/?^  — 7^^ — 2\kh 

+4/^2  = _ Q ^^2 ^ „ Q ^^2^ 


Multiplication  of  Monomials 


§ 1 21.  The  first  exercises  are  to  impress  the  pupil  with  the 
reality  of  meaning  of  algebraic  multiplication  through  the  use 
of  factors  that  are  concrete  numerical  dimensions  of  conceiv- 
able geometrical  figures.  The  areas  are  the  concrete  interpre- 
tations of  the  products.  The  work  called  for  by  the  paragraphs 
in  fine  print  is  by  no  means  waste  of  time.  In  2,3,  and  4,  do 
some  in  class  orally  and  assign  others  for  home  work. 


Answers: 

Problem  i.  (i)  35  sq.  in. 

(2)  225  sq.  cm.  (or  cm^.) 

(3)  7«  sq.  ft. 

(4)  <^\x  sq.  m.  (or  m^) 

(5)  P(l  sq-  rd. 


(6)  sq.  yd. 

(7)  a¥c  sq.  mi. 

(8)  6ys  sq.  km.  (or  km*.) 

(9)  2ox^y^  sq.  ft. 

(10)  i4fa*6*6*  sq.  in. 


74 


First-Year  Mathematics  Manual  [pp,  162- 

Problem  2.  225  sq.  ft.,  672V  sq.  m.,  a^  sq.  cm.,  s6|6V 

sq.  in.,  sq.  in.,  a%^  sq.  mi.,  i2\p\^  sq.  m.,  sq.  ft., 

18.6624^46^^  sq.  dm.,  ^im^n^p^q^  sq.  cm. 

Problem  3.  421I  cu.  cm.,  337J  sq.  cm.;  8:;^^  cu.  ft., 

24a:4sq.ft.,i5iifa363cu.  in.,  sq.  in.,  13.824/  cu.  m., 

34.56/  sq.  m.,  64x3/23  cu.  cm.,  96x^2^  sq.  cm.,  175.616 
xy2^  cu.  mi.,  188.16x4/24  sq.  mi.,  35i||x^^  cu.  rd.,  64f|x« 
sq.  rd.,  1,728^^6^  cu.  yd.,  864  sq.  yd.,  cu.  ft.,  121^^^°  sq. 

ft.,  3,375//^^  cu.  m.,  1,350/^4;^"  sq.  m. 

Problem  4.  (i)  462  sq.  in.,  540  cu.  in. 

(2)  (34f+i6|?x)  sq.  ft.,  i7^x  cu.  ft. 

(3)  i6(x+y)+2xy  sq.  mi.,  8xy  cu.  m. 

(4)  2(a3+a^+a7)  sq.  rd.,  a^  cu.  rd. 

(5)  (4x3+ iox^+ 20x7)  sq.  yd.,  iox^°  cu.  yd. 

(b)  (31.5/+112/+144/)  sq.  cm.,  2$2p^^  cu.  cm. 

(7)  2{x^y^z^+x^y^z^+x^y^z^)  sq.  ft.,  x4y424  cu.  ft. 

(8)  {i2ahp^m^+i6acm^p'^-\-2^bcp^m^)  sq.  in.,  2/^abcm^p^ 
cu.  in. 

(9)  (2ox^^+||x'*®+fx^^)  sq.  mi.,  x3®  cu.  m. 

(10)  (42x33;3-[-4|-x7y^+3|xy)  sq.  cm.,  qxy  cu.  cm. 

Problem  6.  By  adding  exponents  of  the  factors.  No. 

Problem  7.  abc  cu.  ft.,  abc  cu.  ft.,  abc  cu.  ft.  The 

volumes  are  all  the  same.  No  effect  on  value  of  product. 

Problem  8.  ^xyz,  ^xyz,  Txyz,  ^xyz,  49x72,  49x3/2,  49x3/2, 

343xy2. 

Problem  9.  (i)  and  (3),  (5)  and  (7),  (9)  and  (ii). 

Problem  10.  First  and  second  are  equal;  but  third  is 
different  in  value  from  the  others. 

Problem  ii  (see  § 64,  page  84).  A product  is  multiplied 
by  a number,  by  multiplying  any  one  of  the  factors  by  the 
number.  This  is  true  for  both  + and  — factors.  Chan- 
ging the  sign  of  a factor  is  the  same  as  multiplying  the  factor 
by  —I,  and  this  multiplies  the  whole  product  by  — i,  or 
reverses  its  sign.  Changing  the  sign  of  another  factor 


The  Fundamental  Operations 


75 


-165] 

reverses  the  sign  of  the  product  again,  or  brings  it  back  to 
what  it  was  originally.  Hence,  changing  the  sign  of  any  even 
number  of  factors  does  not  alter  the  value  of  the  product;  but 
changing  signs  of  an  odd  number  of  factors  reverses  the  sign 
of  the  product.  Illustrate.  We  may  also  say  an  even  num- 
ber of  negative  factors  gives  a + product  and  an  odd  number 
of  negative  factors  gives  a — product. 

Exercise  XXI 

Have  pupils  ready  to  report  answers  to  from  10  to  15 
of  the  exercises. 

Answers: 


I. 

-3400 

13- 

2. 

14.  — 

3- 

15.  (a+5)« 

4- 

^omH^ 

16.  (r— 

5- 

22ox'^y^z^ 

17. 

6. 

360^757^:7 

18.  (x^+y^)9 

7- 

^2p^q^r^ 

19.  i2{x—yy^{x+yy 

8. 

— 12^.  ^2 ^Ga^b^mW 

20.  {a—by^ 

9- 

— 2^x^p^q^r^z 

21. 

10. 

t^Ss^^U^S 

22.  — 

II. 

— ^X^OylO^II 

23.  —202^p^^ 

12. 

24.  gx^^. 

Division  of  Monomials 


§ 122.  Do  a few  orally. 
18  ft.,  2 ft. 

Problem  2: 

(1)  a=2tit. 

(2)  a=x^  in. 

(3)  6 = |x^ys^m. 

(4)  b = 21.  sd^fc 

(5)  a = gpxH^ 


Problem  i.  16  ft.,  9 ft.,  12  ft., 

(6)  a = ^l^k^c^ 

(7)  b = v^u^c^ 

. (8)  a — o.^h^x^ 

(9)  a = I . \zx^p^ 

(10)  b = ^x^y^z^. 


76 


First- Year  Mathematics  Manual  \pp-  165- 


Problem  3.  Give  the  base  an  exponent  equal  to  the  differ- 
ence of  its  exponents  in  dividend  and  divisor. 

Problem  4.  See  page  86. 


Exercise  XXII 


Assign,  say,  the  even  numbers  for  home  work. 


Answers: 


I. 


2. 

3- 

4- 

5- 

6. 

7- 

8. 


9- 


10. 


II. 


12. 

13- 

14. 

IS- 

16. 

17- 

18. 

19. 


1 . 2,a^x^y^z^ 

-Stu^ 

^a^b^c^ 

-S\P^q^ 

— 3Sa;“yV® 

4a* -W 
1 . 3x"y  V 
—fp^”q”r^” 

— 2{a-\-b) 

4T(x^-fY 
— o.o7(a^+2a6+i*)4 

+3 

2x^y(a^+b^y 
— 4}{a-{-by^{a—by* 
Sx^^y^^{x-\ryy 

— 6p^  {p  -{-q — r)  ^ 

72"* 

005^'^ 

a^b^ 

121 

2s(^+y)'‘(*— y)^’ 


20. 


-1 68]  The  Fundamental  Operations  77 

Multiplication  and  Division  by  Means  of  Exponents 


§ 123,  page  167,  Problem  i.  2^  3^ 

3^  3^  3“ 

210  2^ 

5*  5"  5®- 

Exercise  XXIII 

Do  two  or  three  with  the  class,  and  have  them  verified 
by  actual  multiplication.  Assign  others  for  home  work. 
Answers: 

1-  3*  • 3''=3''  = S3i.44i  (See  table,  p.  167) 

2-  5^  • S''=5"=48,828,i25 
3.  2^-7-24=25  = 32 

4-  3^  • 3^=3"  = i77,i47 

5.  2”^27=24=i6 

6.  3”^3^=3^='243 

7-  5*-^5’=S^  = i2s‘ 

8.  5''-^5^=S^  = 3,i25 

9-  (3^)' =3'“= 59,049 

10.  (5^)^  = 5'^  = 244,140,625 

11.  2'^-i-2*=2®  = 64 

12.  s«  . 5‘'=5'^  = 244, 140,625 

024 

13.  ^ = 34  = 81 
2^20  ^ 

14-  5®  • 5^  • S”=  5'^  = 1 1,920,928,955,078,125 

15.  (2^y  = 2^^=l6,'J'J'J,2l6 

22s  , ^12  ^37 

16.  ^ = ^ = 1 

322  3x5  337 

CIS  t-i8 

17-  -—^^7—=  5'”= 9,765,625 

224  2^0  2^^ 

—,-7^= 2'*= 262,144. 


78  First-Year  Mathematics  Manual  [pp.  i68- 

Multiplication  and  Division  of  a Polynomial  by  a Monomial 

§124.  Problem  i.  ^>(s+^^+‘^+7i+4i+9- 5+/)  and 
. sb-\-by^ 

Problem  2.  a^{a^-\-7,a¥+2>(^^b-\-b^)  and  a^-{-^a^b^-\-^a^b-\- 
a^b^ 

+3^6^  -\-2>^^b  =as  -\-^a^b^  +3^46  + 
a^b^ 

Problem  3.  By  multiplying  each  term  of  the  polynomial 
by  the  monomial  and  connecting  the  resulting  products  with 
the  signs  determined  by  the  law  of  signs  for  multiplication. 

Each  term  of  the  product  is  the  product  of  corresponding 
terms  of  the  polynomial  by  the  given  monomial. 


Exercise  XXIV 

Do  one  with  the  class.  Assign  others  for  home  work. 
Answers: 

1.  — 

2 . 2oa^b — ^oa^b^ + ^oa¥ 

3.  — 3 .6a6x4+6.9^ii;r^y+i2a:4y2  — 2 . ibx^yz 

4.  — . ^a^p^m^ 

5.  1 2^"*+ 1 5^16  — 156^ 

6 . 1 1 — 2 2 — 3 D + 9I&4 

7.  i2x^  — i4x^y+2  2xy^— i2y^ 

8.  -IS -9^" -53^^+176" 

9.  goax—2>S^^ 

10.  — 2oa^  — 1 1 8a6  + 66ur 

11.  4x^  — 72^:^ 

12.  — ioy^-“36y* 

13.  ^ = 48 

14.  :^£:=  — IS 

15.  x=-2 

16.  a = s. 


The  Fundamental  Operations 


79 


-170] 

§ 125.  Do  two  or  three  problems  with  the  class,. 

Answers: 

Problem  i.  x andx’;  x and  and  x and  4^^,  or 
and  x"*;  x^  and  3x7,  and  x^  and  43;^. 

Problem  2.  Sketch  a rectangle  of 

(1)  Base  X (or  4x— 33^)  and  altitude  4X— 37  (or  x). 

(2)  Base  5x  or  x^  — 2X3;+33^%  and  altitude  x^  — 2x7+33;^ 
or  5x,  respectively.  Other  solutions  also  possible. 

(3)  Base  Ta^lfc^  or  26— 3^+5^,  and  altitude  26— 3^+5^; 
or  Ta^b^c^,  respectively.  Other  solutions  also  possible. 

(4)  Base  3m  or  and  altitude  m^—^m^n+ 

2n^  or  3w  respectively.  Other  solutions  also  possible. 

(s)  Base  5x^  or  3x^--2x+i,  and  altitude  3x^  — 2x+i  or 
5x^  respectively.  Other  solutions  also  possible. 

Problem  3.  By  factoring  a monomial  out  of  each  term 
of  the  polynomial:  No;  e.g.,  x^— 3X— 28  cannot. 


Exercise  XXV 

Develop  several  with  the  class.  Assign  others  for  home 
work. 

Answers: 

1.  5{a-2b) 

2.  i7x^(i  — lyx) 

3.  2x{Sx  — ab) 

4.  a(x+y— 2) 

5 . xy'^z^  ( 1 4x — yx^y + 8) 

6.  i5m^;zV(4n— 3wr+6wV+6m^^) 

7.  3a^Z>V(i . $bc^-‘0 . 4ac^— o . ga^c+T^ab^) 

8.  4a^x^(i— 3ax— sa^) 

9.  p{ap^-\-zpx-Jtq-\-i^axq) 

10.  x(x^ — pqxA-p^q^  — pq^x'^) . 


8o 


First-Year  Mathematics  Manual  [pp,  lyo- 


§ 126.  Answers: 

1,  — 76^ 

2.  (i)  s^+s^ 

(2)  4xy^—5y^  — 2x^ 

(3)  p+l-r+t 


(4)  2 

(5)  a+b-c+d 


3.  By  dividing  each  term  of  the  polynomial  by  the 
monomial.  It  is  not. 


Exercise  XXVI 


Answers: 

1.  4a^— 3^6+66* 

2.  — 2x+3:;i£:^y— X* 

3.  — 6y^+7x^yS2; 

4.  sb^—6a^b  — ya’^b+3ci'^b^c^ 

5.  —;^amn^-i-4bm'^n  — ioabmn-\-‘yd^mn 

6.  —5  — gabcp^q^ 

7.  — 66+ 146^  — 1 264 

8.  2 — ioy2; 

9.  sa6c3— 3a^  — 1 5^3+7^36^:— 

I o.  --^bdqs + ^bcqt+6cdts. 


Exercise  XXVII 


Answers: 

1.  2>(^^bc^2ab^c+^CLbc^ 

2 . -|xSy 7 — 1 4x^y^  +21  x^y —31  Jx7y7 

3.  —6oa^b^c-\-^ooa^b^c-]r2>^ci'^b  c^  — T^a^b^c^ 

4 . 5 ab^c^d^  ( 5 a^b^c^d^ — 3 + 6^/7) 

5.  7xy2(3x4y^2^4_ 2xS2^3+5y92”  — 6x^y'4) 

6.  (:«;-y)(4a-36+4c) 

7.  5x“y2'^— 

8.  -3(^+3')"+S(^+>')®+7 

9-  — 3/’(^"+^")(>'+^)+6/'(^^+^^)’+i 


The  Fundamental  Operations 


8i 


-173] 


6 

M 

10 

11 

<0 

II. 

c=-A 

12. 

II 

1 

M 

0 

Multiplication  of  Polynomials 

Refer  the  class  to  Fig.  22,  page  24.  Do  i,  2,  and  3 
orally,  and  assign  the  rest  for  home  work. 

Answers: 

1 . (x+y)  {a+b)  = ax-\-bx-\-(iy-\-by 

2.  (i)  CL(^cA~d)A~b{c-\~d) 

(2)  Q,c-\~(^d-\~bc~]rhd 

3.  (i)  (a+6)c+(a+6)c? 

(2)  Q,c-{~bc-\~(i'd-\~bc 

4-  (i)  3(^+2)+4(6+2);  3 • 6+3  • 2+4  • 6+4  • 2 = 56 

(2)  5(^+3)+^(^+3);  5^+3  • S+^^'+3^  = ^'+8a+is 

(3)  {a-\-b)a  — {a—6)2\  a^+6a  — 2a  — i2  = a^+4a— 12 

(4)  {(i-\~b)b-\~{,(i'~\~b)c]  cib-\~b^~\~ci(^~\~bc 

(5)  i^+y)y+(.oc+y)z;  xy+y^+xz+yz 

(6)  {Q>-{~b)(i^~{~{(^~{~b)b^-\~{^~{~b)2aby  ci^ -\~ci'^b-\~(^b^ 

6^+ 2a^6+2a6'*  = a^+3a^6+3a6^+6^ 

(7)  (x--y)x^+  {x—y)  2xy+  (x— y)y^ ; x^— x^y+  2x^y— 

2xy^ +xy  ^ — ^3  = ^3 — ^2^  _j_  ^y2  __  ^3 

(8)  {a—b)a^—2ab{a—b)A-{(i—b)b'^\  a^—a^b—2a^b+ 

2ab^-\-(ib^  ^b^  = — 2>^'^b-\-2)(ib^ —b^ 

(9)  (x^  — 2xy  +y^)x  + (x^  — 2xy  +y^)y  i “ 2x^y  + 

xy^+x^y  — 2xy^+y^  = x^ — x^y — xy^+y^ 

(10)  (S+^)2S  + (S  +x)iox  +(s  +x)x";  125  +2Sx  + 
5ox+iox^+5x^+x^=  i25+75x+isx^+x^ 

5.  Multiply  every  term  of  one  polynomial  by  every 
term  of  the  other,  and  write  the  several  products,  each  with 


82 


First-Year  Mathematics  Manual  [pp, 


its  resulting  sign,  as  a polynomial.  Simplify  the  resulting 
polynomial. 

§ 128.  I.  See  answers  to  4 above.  Discuss  i and  2 
with  the  class. 


Exercise  XXVIII 

1.  a^x^  — ¥y'^ 

2 . a^-\-  a^lf +2^6^+ a^h + ¥ 

3-  P^-p"~5p-3 

4.  ;^oa^—4oa^b^  — i^a^b^-\-2ob^ 

5 . 18. 4a^bc^  — 18. 2a^b'^c — 6a^b^ 

6.  a^-{-a^b  — ab^—b^ 

7 . 1 6x^+Sx^y — 2xy^ — y^ 

8.  p^+p^r^+r^ 

9.  8ik^-\-gkH^-\-t'^ 

10.  lom^  — 38^^5+51^5^—30^^ 

11.  — 9 + 5 + 1 5 — 2 5 + 1 2 — 1 6ab^c + 

8a^c  — 1 8abc^ + 24a6^  — 1 2 

12.  81  x4+  9xy  + 1 8xy^ +4y4 

13.  3ooa^6+i66^ 

14.  4+— 24/A  — 9/^^ 


§ 129.  Develop  some  with  the  class,  and  assign  others. 
Answers: 


1.  m"*- 2wx+r 

a^+2a6+6^ 

2.  :\;^+2/?x+/>^ 
z)^+2ai;+a^ 
x"*  — 2/?X+/?^ 


p^  — 2pk+k^ 
p^-\-2pk+k^ 
k^  — 2kr-{-r^ 

r^+2ry+y^ 

k^+^kn+n^ 


X*  — 2xy+y^ 
m^+2mx+x^ 

— 2ry+y^ 

h^-2hp+p'‘ 


3.  Write  the  square  of  the  first  term,  plus,  or  minus,  the 
double  product  of  the  two  terms  (according  as  the  given 
binomial  is  a sum  or  a difference),  plus  the  square  of  the  second 
term. 


The  Fundamental  Operations 
Exercise  XXIX 


83 


-175] 


Answers: 

1.  x^+iojt;+25 

2.  /-i43;+49 

3.  16a'*  — 8a+i 

4.  lxf+ixy+\y* 

5.  .i6a^—  .24^/+.  09/* 

6.  — 24/>^X3;+ 1 

7.  .36.Ty2^+i  • 2x3/2+! 

8.  (a+6)^+6(a+6)+9 

9.  (a— 6)''+2c(a— 6)+c^ 

10.  (x-3/)"-|(x-3/)+i| 

11.  {k+iy-2>^{k+l)A-2.2Ss^ 

12.  (3a-46)"-4c;(3a-46)+4c* 

13.  (ax + 63/)  ^ — 2 ^2  ( ax + 63/)  + c^2* 

14*  {sp(i-^kiy-^Kspq-2>k^)+^^P^i^ 

15.  i 2\ab^c — f a6(;^)^+  i4a^6c(2^a6'*^;  — ^abc^)  +49a46V 

16.  (a+6)^+2(a+6) (c — J)  + (c — 

17.  iia^+146^ 

18.  (4/-5x)^+2(4/-5x)(55-43/)  + (5.9-43;)^ 


§ 130.  Answers: 


a^—x^ 

f-s^ 

r^  — 16 

f-h^ 

225—2^ 

k^-h^ 

i^a^—gb^ 

45"*-- 169 

i.//)2  1 y2 

aP  9' 

.ogxH^—  .o4y^k‘ 

P-u^ 

a^b^c^—x^y^z^ 

3.  The  product  of  the  sum  by  the  difference  of  the  same 
two  numbers  is  the  difference  of  their  squares. 


84 


First-Year  Mathematics  Manual  [pp.  iy6- 


Exercise  XXX 


Discuss  the  best  combinations  of  factors. 
Answers: 


I. 

40.^— gb^ 

14. 

k*-i* 

2. 

9Jif — 162^ 

IS- 

f-f 

3- 

16. 

i—p^ 

4- 

ir^-¥" 

17- 

{$x^  — 2y^y—x^y‘ 

5- 

18. 

6. 

i6a^6V— 9 

19. 

7- 

i-2Spy 

20. 

^2a^y2S 

8. 

I— a4 

21. 

4m^‘—gs^ 

9- 

^^4  — 2;4 

22. 

{a+by  — {c+dy 

10. 

(a+6)^-c^ 

23. 

(x^—yy—{x^-yy^y 

II. 

{e^by—h^ 

24. 

{i2ab+\bcy — ( . 4ac — 2b'^y 

12. 

(jn+nY—t^ 

25- 

Ihx^—h^y — i^x^+h^y. 

13- 

(2a— 6)^— 95^ 

Exercise  XXXI 

Answers: 

1 . — 6a^ — Sa^b^ + 6a^b^ 

2. 

3. 

4 . — 4a(;  + 2 — 46^: 

5 . 4a^ + 96^ + + 1 2 — 4ac; — 66c 

6.  a4-|-54-[-9^;6 — 6aV+ 2a'*6^  — 66^c^ 

7 . 4^4 + 4a''6'* + 64 — 9^4 

8.  sx^+8y  — 92"*  — 14x3/+ 1 2x2  — 63^2 

9.  iSm^n 

10.  93^4  — 1 3 5x3;^ + + 2 2 5x^3^ — 30x4 

1 1 . a'^x^ +2^6x3;+ by^ — 

1 2 . a^x^ + b^y^ + — 2 acxz + 2 abxy — 2 bcyz 

1 3 . a^x^ + 6^3^^ + — 2abxy  — 2 acxz + 2 bcyz 

14.  aV+6y +cV  — 2a6x3^+  2acxz  — 26cy2 


-178] 


The  Fundamental  Operations 


8S 


15.  2 m^s — 4 \st^ — ^m^t  — i2mt^ 

16.  —o . 28/4+0 . gmP-\-2  .S2mH^—o.  6^mH—o.  ogm^ 

17.  —^abxy— ^bcy  z 

18.  a^+;^a^b+^ab^-^b^ 

19.  a^  — T,a^b-\-^ab^  — b^ 

20.  8:r^  — i2:s[;^3;+6xy — y 

21.  x^-\-6xy-\-i2xy^-\-Sy^ 

22.  2i6::i;^}'+i28}'^ 

23. 

24.  — o.  25:^^  — 1 . 2xy-{-o . 7,6y^ 

25.  i6p^q—^p^q^-{-^p^q^-{-6p^q^. 


Division  of  Polynomials 


Answers:  i.  ax+ayA-bx+by+cxA-cy]  ax-{-(^y]  hxA- 
by;  cx-{-cy. 

From  the  first  terms.  Develop  orally  with  the  class. 

2.  The  factor  which  multiplied  by  x+y,  gives  the  pro- 
duct ax+dy+bx+byA-cx+cy, 

By  dividing  first  term  of  dividend  by  first  term  of  divisor, 
ax-^x  = a,  first  term  of  quotient. 

By  multiplying  the  quotient  term  a by  the  whole  divisor, 
^+3^?  ax-{-ay.  Then  ax-\-ay-\-bx-\-by-\-cx-\-cy  — 

{ax-\‘dy)=bx+by+cx-\-cy,  the  remainder. 

3.  bx-^x  = b;  and  bx+by+cxA-cy—{bx-\-by)=cx+cy^ 
the  remainder.  cx-i-x  = Cy  and  cx+^:y— •c(:r+y)  =0,  where- 
upon the  whole  quotient  is  a-\-b-\-c. 

4.  Examine  the  form  on  page  178  closely. 

5.  a^+2a6-f-6*  quotient. 


a+b)a^+2>(Fb+2,db^+b^ 

2a^6+3a6^ 

2a^b^2ab'^ 


-\~d^  • d — \ the  first  term  of 
a^(d+b)  ) quotient 


A-2a^b-^a  = -\-^db 
= 2ab(a-\-b) 


) second 
> term  of 
) quotient 


alr‘-\-b^  -\-ab^-i-a=  } the  third  term 

ab^+b^  +b^{a+b)  [ of  quotient 


86 


First-Year  Mathematics  Manual  [pp.  lyS- 


Then  checking:  (a+6)  {a^-\-2ab-\-b^)  = a^-]-2a^b-\-ab^-i-ci^b-\- 
2ab^-\-b  = 

Answers  to  questions  of  5,  page  178: 

1.  The  first  term  of  the  dividend  divided  by  the  first 
term  of  the  divisor  +a3-^+a  = ^-a^ 

2.  By  multiplying  the  whole  divisor  by  the  first  term 
(+a^)  of  the  quotient. 

3.  The  first  term  of  the  remainder  divided  by  the  first 
term  of  the  divisor  (+2a^6-^a  = +2^z6). 

4.  By  multiplying  the  whole  divisor  by  the  second  term 
(+2a6)  of  the  quotient. 

Exercise  XXXII 

Have  the  pupil  solve  the  following  by  actual  division,  and 
test  by  multiplying.  If  he  happens  to  discover  a law  of 
factoring  by  himself  let  him  use  it,  but  do  not  make  this 
the  objective  point. 

Quotients: 

1.  3/+4<y;  Test:  (3/+45)(3/+4^)=9/"+245^+i65" 

2.  0.3X— o.4y;  Test:  (0.3X— o.4y)(4X+7y)  =i . 2X^  + 
o.sooy—2.^y'^ 

3-  3^''  — S^^+b/%  test  by  multiplying  it  by  2k  — ^t 

4.  Arrange  dividend  thus:  iox^—/^x^y  — $xy^  — $^y^\ 
quotient  iox^+i6:x:y+27y'*;  test  by  multiplying. 

5.  9^^  — i2a&+46^.  Test 

6.  3a  — 26.  Test 

7.  Test 

8.  9/^+125^+165^.  Test 

9.  Sa^-\-b^‘  Test 

TO.  Arrange  thus:  — 25x4+i5x3y+iox^y^  — 9:ry^+3y4; 
quotient:  — sx''+3xy— +.  Test 

II.  2jr— y.  Test 


The  Fundamental  Operations 


S7 


-i8o] 

12.  i6a^-\-^a^b^-\-b^.  Test 

13.  o .o^sH^—o  .2stv-\-v^ 

14.  a^+b^+c^ 

IS- 

16.  2x^-{-T^y^ 

17.  0.2St-\-V 

18. 

19.  a-\-b 

20.  3/^ — 25^ 

21.  ^m^  — 2n^ 

22.  4x^  — 5^^ 

23.  9^4— 6m^/^^+4m4 

24.  i6w4-j-2 .4wV+o.o9z;^ 

25.  9 w4 + 6 + 4^4 . 

Summary 

Read  the  summary  through  with  the  class,  and  see  that 
everything  is  understood.  It  is  a good  plan  to  have  indi- 
vidual pupils  read  one  paragraph,  and  then  tell  what  it 
means,  or  illustrate  its  meaning  by  an  example  of  their 
own  making.  Do  not  conclude  too  readily  that  pupils 
really  understand  what  they  read  glibly.  They  may  even 
recite  verbal  definitions  without  a halt,  and  have  no  real 
conception  of  their  meaning.  The  best  test  always  avail- 
able is:  Can  they  give  a correct  illustration  on  demand? 
After  seeing  that  the  statements  are  understood  by  most 
of  the  class,  assign  the  page  as  home  work  to  be  learned. 


CHAPTER  IX 


PRACTICE  IN  ALGEBRAIC  LANGUAGE,  GENERAL 
ARITHMETIC 

The  fundamental  algebraic  operations  have  now  been 
carefully  taught  and  applied,  the  equation  has  been  used 
extensively  under  common-sense  modes  of  treatment,  and 
enough  reconnoitering  has  been  done  to  make  an  explicit 
study  of  the  equation  profitable.  But  unless  the  equation 
can  be  felt  to  state  real  number  relationships,  manipulating 
it  can  give  only  mechanical  training,  eminently  desirable 
when  it  is  clearly  understood  what  the  training  is  about  and 
why  it  is  worth  while,  but  of  almost  no  value  as  unmeaning 
or  blind  exercise.  Modes  of  manipulating  equations  ought 
to  mean  kinds  of  thinking,  and  the  surest  way  for  the  pupil 
to  realize  this  is  to  have  him  acquire  enough  mastery  of  the 
language  phase  of  the  equation  to  enable  him  to  seize  the 
thought  expressed  in  algebraic  language  and  to  clothe 
quantitative  thought  in  this  language.  No  one  would 
expect  to  reason  with  precision  upon  ideas  that  had  to  be 
acquired  through  an  unknown  language  and  no  one  would 
expect  to  profit  by  a technical  study  of  the  laws  of  use  of  a 
foreign  tongue  without  first  having  some  appreciating  sense 
of  its  expressive  power.  With  foreign  tongues  we  require 
considerable  translating  of  simple  language  forms  both  into 
and  out  of  the  foreign  tongue  before  taking  up  the  philology 
of  the  tongue.  Furthermore,  experimental  studies  of  the 
sources  of  unsatisfactory  results  in  algebra  trace  most  of  the 
bad  results  to  failures  in  thinking,  and  in  his  work  with  the 
equation,  to  the  inability  to  translate  common  language 
into  algebraic  language,  and  the  reverse.  It  would  seem  then 

88 


-i8i\  Practice  in  Algebraic  Language  8g 

that  to  get  the  laws  for  manipulating  equations  really  under- 
stood^ some  of  the  difficulties  on  the  language  side,  i.e.,  in 
setting  up  the  equation,  may  well  be  grappled  with  and 
gotten  out  of  the  way  at  the  outset  of  the  intensive  study 
of  the  equation. 

The  work  of  this  chapter  should  be  assigned  about  in 
accordance  with  the  following  subdivisions.  If  the  stipu- 
lated lesson  for  a given  day  is  thought  too  heavy  it  will  be 
best  to  omit  certain  problems  that  are  thought  to  be  of  lessei 
value,  than  to  shorten  the  assignment  and  put  more  than 
this  amount  of  time  on  the  chapter.  Movement  and  con- 
nection of  steps  are  here  of  all  importance,  and  connections 
are  very  materially  aided  by  moving  on  to  the  next  step 
before  the  impression  of  the  preceding  is  lost,  or  too  greatly 
enfeebled. 

First  lesson:  pages  181-84  inclusive. 

Second  lesson:  pages  185-87  inclusive. 

Third  lesson:  pages  188-92  inclusive. 

Fourth  lesson:  pages  193-97  inclusive. 

Fifth  lesson:  pages  198  to  § 143,  page  200. 

Sixth  lesson:  § 143,  page  200,  including  most  of  the 
exercises  to  page  107 — Summary. 

Seventh  lesson:  Summary  and  review. 

Translating  Verbal  into  Symbolic  Language 

The  work  of  the  28  exercises  of  pages  181-83  should  be 
gone  over  first  orally  with  the  class,  and  then  the  exercises 
should  be  assigned  as  home  work,  the  translations  called 
for  being  written  in  notebooks,  brought  to  class,  the  note- 
books interchanged  among  the  pupils,  criticized  and  cor- 
rected by  pupils,  and  then  they  should  be  gone  over  carefully 
by  the  teacher. 


90 


First-Year  Mathematics  Manual  [pp,  i8i- 


§132.  Answers  to  exercises: 


I. 

16,  17,  18 

16,  IS,  14 

2. 

n,  w+i,  nA-2 

n,  n—i,  n—2 

3- 

2X,  2X+2,  2X+A 

2X,  2X  — 2,  2X  — ^ 

4- 

w+3>  «+5)  ^+7 
w+3,  w+i,  w— I 

S- 

^+^+3+^+6 

6. 

i+;^+/z+i  =3^ 

7- 

W+W+I-4-«+2  = 2I 

3W+3  = 2i;  w=6,  7,  8 

8. 

w(;^+l)  = 72 

9- 

/z+/z+3+;^+6  = 27 

37^+9  = 27;  w = 6,  9,  12 

10. 

4X10 

II. 

12. 

2WX  lO+W 

13- 

(i)  — 12 

(7) 

2,X—2{X'-6) 

(2)  X+12 

(8) 

60  — X 

(3)  2:v 

(9) 

x—60 

(4)  4(:i:-8) 

(10) 

x-\-hx 

(S)  6(:»:+3) 

(ii) 

X— 3(x— 20) 

(6)  |(x-i5) 

(12) 

X : X+14 

14. 

IS- 

12X 

16. 

X 

12 

17- 

10/+/-3  = io(/-3)+/+27 

18. 

2(«+25)_« 

8 “2 

Practice  in  Algebraic  Language 


-184] 


19. 


20. 


21. 


22. 

23- 

24. 

26. 


2;z+3(n+4) 

2;^— 3(/^— 4) 
2n—;^{n—2) 

2/^+3(^“4)  = i3 
72,  48,  o,  -24,  o,  10 

lfr+4),^, 


25—11 


1(83;- 20), 


2(83;— 20) 


5(2^+9) 


, |(2X+9). 


91 


§ 133.  A treatment  similar  to  that  suggested  above 
should  be  given  to  the  49  exercises  of  § 133,  which  revises 
and  formalizes  all  the  rules  of  arithmetic.  Article  132  may 
well  take  a day  in  class  and  another  day  for  revision  and 
recitation  on  these  exercises  as  prepared  work.  Article  133 
should  take  about  the  same  amount  of  recitation  time. 


§ 133.  Answers: 

1.  7+5  = 12 

2.  4 . 10+5  = 45 

3.  6 • 10+8  = 68 

4-  5o+7>  48+9,  18+X 

5.  x+S,  y+15,  x+16,  x+y 

6.  5 = 4a+io 

7.  18  — 7 = 11 

8.  m—s  = d 


9.  sa  = sx 

10.  M • m = P 

11.  m = dA-s 


92 


First-Year  Mathematics  Manual  [pp.  184- 


12, 


13- 

14. 

IS* 

16. 

17* 

18. 

19. 

20. 


21. 


22. 

23* 

24. 

25* 

26. 

27. 

28. 

29. 
30* 
31* 
32. 


33* 


34. 


35* 

divided 


Plm  = M;  = ^ 

Dld  = qj  or  D=d  • q 
D = d • q+r 
M = m-\-b 
P = b+g 
s = cd 
W = slf 
M=bt+r 
S ^ n^  j di  “I”  ^2  / ^2 

D Ui  / di  / ^2 

P “ X ^2  / ^2 

^ ^ /Z'l  / / £^2 

R = ab 
A=s^ 

D = bs 
J'  = T'*0-+Tf7 

/ , h 

v= — I 

10  100 

a , J , c 

z>  = 1 

10  100  1000 

I I I ^2  I 

z;  = iooac+ loa^+a^H 

^ 10  100 

D=  10002+-^ 

100 

$12.60,  31.2  A,  39  mi.,^yd. 

/)=$22.so,/>  = 75  bu.,/>  = 27omen,/>=— A , />=— 

4 100 

6 

r 100 

loop  . . , 

— ^ = r;  rate  per  cent  is  100  times  the  percentage 

by  base. 


Practice  in  Algebraic  Language 


93 


-i88] 


2,880 

26.  — ; — =r  = 8 
360 

3,000 

37.  =r  = 4 

725 


725 


38.  b = 


loop 


39.  i = = 540 

40.  6 = = 

41.  ioop  = br 

42.  7,740 

43.  $3.90,  $7.80,  $11.70,  $19.50, 

<X$3.90 


44.  $3.20^, 


^.8o;,  $3.60^,  $4.48^, 


9.75,  $14,621, 
$8o/r 


100 


45- 


2ort  i35r/  prt 

100  ’ 100  ’ 100  ’ 100 


. prt 
46.  ^ = - — 
100 

looi 

«■  -if-f 


48. 


looi 

pr 


49.  r 


looi 

TT* 


Graphing  Percentage  and  Interest 
page  188 

Work  Exercise  I orally. 

Work  exercises  with  the  class,  making  clear  the  meaning 
of  the  points  and  lines  of  the  graph.  Let  the  teacher  sketch 
3 rapidly  on  the  blackboard.  Let  the  class  solve  4 and  5 


94 


First-Year  Mathematics  Manual  [pp.  i8g- 


anci  have  6 and  7 answered  orally.  Work  the  rest  of  this 
list  through  with  the  class  somewhat  as  a chalk-talk,  getting 
the  class  to  participate  as  much  as  possible. 

Treat  §§  136-37-38  in  the  same  general  manner.  Do 
not  assign  this  work  as  a whole  for  home  work.  At  most, 
only  particular  exercises  should  be  so  assigned.  The  teacher 
should  guide  and  keep  the  class  moving  right  along.  Time 
may  be  easily  wasted  by  treating  such  work  by  mechanical 
assignments. 

§ 135.  Answers  to  some  problems: 

I.  $2,  $4j  $6 

6.  Zero  % of  any  base  is  zero  dollars. 

7.  Between  them. 

9*  P~Th^^ 

10.  p=i,b-,  p=^i-,b;  p=^b 

11.  2%  of  $250;  4%  of  $250;  2%  of  $150;  4%  of  $150 

12.  By  reading  length  of  E C,  etc. 

14.  By  reading  lengths  of  appropriate  verticals  from 
O X up  to  the  graph. 

18.  p = $iXr;  p = $4r 

19.  Percentage  of  any  base  at  o per  cent  is  $0. 

§ 136.  Answers: 

I.  $3,  $6,  $9,  $12 

7.  Int.  for  o yr.  on  any  base  at  any  rate  is  $0. 

8.  Lengthening  of  verticals  proportional  to  correspond- 
ing increases  in  lengths  of  time-line. 

The  Evaluation  of  Expressions 

The  first  4 exercises,  page  195,  should  be  gone  over 
with  the  class  orally  and  rapidly.  Exercises  5 and  6 may  be 
assigned  for  home  work  and  the  rest  should  be  worked  out 
with  the  class  as  a laboratory  exercise  in  the  classroom. 


~ig8\  Practice  m Algebraic  Language  95 


I. 

D=i^o  ft.,  0 = 85  mi. 

2. 

^=312  sq.  ft.,  ^=35-7  sq.  in. 

3- 

^1=96  sq.  ft.,  ^=31.898  sq.  rd. 

4- 

44=532,  44  = 237.98 

5- 

(i)  800 

(6) 

9 

(2)  3>75o 

(7) 

151 

(3)  1,170 

(8) 

950 

(4)  2 

(9) 

1,098 

(s)  798 

(10) 

1,800 

6. 

</,  = 2i|  ft.,  (/i  = 4o|  in.,  di  = 45 

in.,  d 

, = i8f|  in.; 

31.5  lb.,  Wi  = 25.5lb.,  W2  = i2/iflb. 

7- 

2£;=i,036.8  lb.;  v = 4o  cu.  in.;  d = 

5 lb. 

8. 

h = io  rd. 

9- 

5 = 256  ft.,  y = 2,ii6ft.;  ^ = 2 sec.. 

, ^ = io  sec. 

10.  ^ = 2,340 ft.,  5 =1,080 ft.;  z;  = 20 ft.  per  sec.,  z;  = i6 
ft.  per  sec. 

11.  /=V-sec.,  /=!  sec.,  /=ifi-ft.,  / = i6Xffrft. 

12.  A = 6,  (5=6),  ^=30,  (5  = 15) 

13.  = 

14.  /2=4y^,  72=6f|^ 

15.  £ = 150,  £=44s|;  M=i;  M=4 


16.  V- 

§ 140. 

I.  (i) 

= 84,  F=74. 

Answers: 

C = 66  ft., 

d = 28  ft. 

(2) 

C=22  ft.. 

d = 2i  ft. 

(3) 

c=-v  ft.. 

<^  = 5i‘r  ft. 

2.  (i) 

C=i8f  ft.. 

r=2i  in. 

(2) 

C=37f  ft.. 

r=io|  ft. 

(3) 

C=iis\  ft.. 

r=ij^  rd. 

3-  (i) 

.4  =28|-  sq.  ft.. 

r = 7 ft. 

(2) 

21  = 154  sq.  ft.. 

r=|/7o  ft. 

(3)  ^ = 1,386  sq.  ft.,  r = 3'Vi/2,3io  ft. 

Assign  Exercises  4 and  5 for  home  work  to  be  brought  to 
class. 


96 


First-Year  Mathematics  Manual  [pp,  ig8- 


4.  (i)  7=179!  cu.  ft.,  r=i  in. 

(2)  F=5,S77H- cu.  ft.,  r=2  ft. 

(3)  ^=33if  cu.  ft.,  r=i%f  4,851  rd. 

5.  C = 6f  yd.;  4=3|sq.  yd.;  5=121  sq.  yd.;  F=4^j-cu. 
yd. 

21  = 2544-;  5=i,oi8f;  F = 3_,054f;  r = g 
C=37f;  5=452f;  F=9o54;  r=6 

C=3if;  A = ^^■,  7=523^;  r=5 

C=i8f;  ^ = 28f;  5=1134;  ^=3. 

Work  through  § 141  orally  with  the  class. 

§ 142.  Answers: 

1.  10 

2.  20  ft. 

3.  24  ft. 

4.  21  ft.  and  28  ft. 

5.  I2jt. 

6.  1/72  or  67/2 

7- 


8.  ^=i5ft. 


10.  9+48+37/ 265=57+31-^265 

I,, 

12.  81  ft. 

13.  VJ^^+t. 


The  Circle 

§ 143.  Read  carefully  with  the  class  the  text  of  pages 
200  and  201  and  work  with  the  class  the  exercises  of  page 


201. 


-204] 


Practice  in  Algebraic  Language 


97 


Answers:  i.  2^=90°,  3a; =90°,  x=3o°,  and  3x=6o°. 


A<4r^ 

\~-2r- 


2. 


7^ 

LJ 


3- 


A >the  square  = 2^"* 


The  Triangle 

§ 144.  See  that  the  class  read  the  first  paragraph  of 
§ 144  understandingly. 

Answers: 

1.  AB  = 68.s  rd. 

2,  3,  and  4.  These  answers  are  to  be  given  from  intui- 
tion. The  thought  is  that  the  exercise  will  focus  attention 
on  what  is  to  be  proved. 

§ 145.  After  the  observational  attempts  of  Exercises 
2,3,  and  4,  the  wording  of  the  theorem  will  easily  be  grasped. 
The  teacher  will  notice  that  this  is  the  first  attempt  at  a 
deductive  proof  of  a geometric  truth.  The  aim  has  been  in 
this  attempt  to  use  the  simplest  possible  untechnical  language. 
The  proof  may  need  to  be  gone  over  two  or  three  times. 
Exercises  1-9,  pages  203,  204  should  all  be  solved  that  the 
proved  truth  may  be  impressed  upon  the  class.  Have  the 
class  point  out  how  the  theorem  applies  in  each  of  the  exer- 
cises. 

Answers: 

1.  c = c']  and  /.B=  ZB' 

2.  a = a',  ZB=ZB',  and  ZC=  ZC' 

3.  The  proofs  are  citations  of  the  theorem  § 145. 

4.  Let  the  pupils  show  from  Fig.  212  how  the  theorem 
applies  and  the  consequences  of  its  application. 

3-9.  Rernark  of  4 applies  here  in  each  exercise. 


98 


First-Year  Mathematics  Manual  [pp,  204- 


§ 146.  Begin  this  section  orally  with  the  class,  having 
individual  pupils  read  and  answer  as  far  as  they  can,  then 
calling  on  other  pupils  to  suggest  and  to  help  on  difficulties. 


§ 146.  Answers  to  exercises: 

6.  B,  C,  D,  E,  and  F 

7.  6 chords 

8.  By  stepping  the  circumference 

9-  ^ ri/3,  r 

10.  361/3 


12.  1501/3 
2 


14. 

IS- 

16. 

17- 

18. 

19. 


20. 


If- 1 58 


/ = 2|/  lO"*  — 6^=16 
l=‘2V  E}—r^ 

5 = -(Ji  + ^>2  + 63l ) 

2 

The  circumference 

rC 

S in  18  becomes  A = — . 


Summary 

Work  over  the  summary  orally  with  the  class,  see  that 
all  the  important  ideas  and  definitions  are  understood,  then 
assign  it  as  home  work,  reciting  next  day  upon  the  important 
statements  and  definitions. 


CHAPTER  X 


-2I0] 


THE  SIMPLE  EQUATION  IN  ONE  UNKNOWN 

Have  the  class  bring  to  the  classroom  a notebook  with 
squared  paper,  and  a small  ruler  for  use  in  laboratory  work. 

Lesson  i 

From  beginning  of  chapter  to  Art.  152,  page  214,  treated 
as  follows: 

Show  on  the  board  the  construction  of  the  graph  of 
Art.  148.  Show  that  the  graph  represents  or  pictures  3/  for 
all  values  of  t,  and  in  the  sense  that  a definite  line  is  either 
given  or  may  be  easily  drawn  to  represent  3/  for  every  value 
of  t that  may  be  chosen.  Show  also  how  to  graph  3^+3 
and  again  in  what  sense  the  line  or  graph  pictures  the  expres- 
sion as  dependent  upon  the  t.  Too  much  talk  about  the 
graph  will  not  explain  but  rather  confuse  the  ideas  of  pupils. 
By  means  of  questions — not  too  leading — have  members  of 
the  class  suggest  and  assist  you  in  making  the  essential  ideas 
clear.  Fifteen  or  twenty  minutes  is  enough  time  to  spend 
on  both  graphs.  Have  the  work  of  Art.  149  done  as  labora- 
tory work  during  this  same  recitation  exercise — everyone 
drawing  all  the  graphs  with  enough  care  to  get  the  notion 
of  the  meaning  of  the  graphs  as  pictures  of  the  polynomials. 

If  the  six  exercises  of  Problem  2 are  not  completed  assign 
the  rest  as  home  work,  the  graphs  to  be  brought  to  class 
next  day.  Assign  also  Arts.  150  and  151  to  be  recited  on 
the  next  day,  completing  in  the  exercise  period  the  answers 
called  for  to  Exercises  1-16,  page  214,  at  least  half  being 
answered  orally.  Go  carefully  through  with  the  class  the 
ideas  of  pages  212  and  213  in  the  recitation  period.  Art. 


99 


lOO 


First-Year  Mathematics  Manual  [pp.  2ij- 


151.1  and  II,  should  be  read  aloud  by  the  teacher  or  members 
of  the  class  and  commented  on  until  understood  by  the  class. 

Lesson  2:  Arts.  152  and  153 

One  of  the  main  sources  of  difficulty  with  first-year 
classes  is  due  to  their  general  inability  to  get  the  sense  from 
written  sentences.  Have  different  pupils  read  to  the  class 
the  several  sentences  of  the  portion  of  the  text  of  Art.  152 
that  precedes  the  exercises,  and  tell  what  the  sentences  mean. 
Work  through  the  exercises  of  page  215  with  the  class,  show- 
ing the  application  of  the  suggestions  for  stating  the  equations. 
Have  pupils  do  as  much  as  possible  of  the  work  of  writing 
the  equations.  Have  members  of  class  read  in  turn  Art. 

152. 1 and  II,  and  explain  them,  with  teacher’s  help  if  needed. 
Read  with  the  class  and  illustrate  the  sense  of  the  five  axioms 
of  Art.  153.  Assign  any  remaining  exercises  of  the  ten  of 
Art.  152  and  the  first  six  of  Art.  154  for  home  work.  Go 
quickly  over  this  work  the  next  day. 

For  answers  see  the  end  of  this  chapter,  pages  106  et  seq., 
of  this  Manual. 


Lesson  3:  Arts.  154,  155,  and  the  first  10  Exercises  oj 
Art.  156 

In  Art.  154  emphasize  the  application  of  the  axioms  of 
Art.  153,  using  the  names  in  ( ).  Finish  the  unassigned 

exercises  of  Art.  154  orally  in  class.  Have  Art.  155  read 
in  class,  and  its  substance  explained,  if  necessary.  Go 
through  the  solution  of  Exercise  i.  Art.  156,  and  if  time 
allows,  solve  a few — half  a dozen  or  so — of  the  exercises 
with  the  class.  For  answers  see  the  end  of  this  chapter, 
page  106  of  this  Manual.  Assign  15  or  20  exercises  of  the 
list,  page  219,  for  home  work.  Do  not  slight  Art.  154. 


Simple  Equation  in  One  U nknown 


lOI 


-220\ 


Reasons,  not  rules  nor  authority,  must  govern  in  mathematics. 
In  Art.  155  make  the  meaning  of  root  oi  an  equation  quite 
clear. 


Lesson  4 

Assign  for  home  work  even-numbered  exercises  from  20 
to  35,  pages  219  and  220,  and  the  problems  of  Art.  157  to 
13,  page  221. 

After  clearing  up  in  the  class  any  difficulties  revealed  by 
the  home  work  of  Lesson  3,  perhaps  explaining  by  solving 
any  exercises  of  pages  219  and  220  that  have  not  been  and 
are  not  to  be  assigned,  work  through  with  the  class — pupils 
giving  such  suggestions  as  they  can — or  with  a pupil  at  the 
board,  question  through  the  solutions  of  the  first  half-dozen 
problems  in  the  recitation  period.  Do  not  allow  the  work 
to  drag,  nor  to  be  diverted  by  too  many  pointless  queries 
and  suggestions.  To  explain  by  telling  over  and  over  as  often 
as  asked  for  the  same  difficulty  encourages  mind  wandering 
and  premiums  inattention  on  the  part  of  the  class.  One 
or  two  repetitions  of  the  same  explanation  with  the  under- 
standing on  the  part  of  the  class  that  these  repetitions  are 
to  suffice,  save  in  exceptional  cases,  will  be  found  more 
effective  than  over-explanation.  Let  pupils  understand 
that  when  explanations  are  being  given  all  that  need  help  are 
expected  to  give  attention. 

For  answers,  see  end  of  this  chapter,  pages  106  et  seq.,  of 
this  Manual, 

Notes  and  suggestions  on  Problems  1-13,  pages  220  and  221 

2.  Let  X denote  the  altitude;  then  x+5  denotes  the 
base,  and  |:r(:r+s)  = K^+S)^~"2o.  Whence  a:  = 3 and  x+ 
5 = 8. 

4.  Let  X denote  the  number  of  seconds,  then  160—30:  = 
1 1 2 — 2X.  Whence  0:  = 48. 


102 


First-Year  Mathematics  Manual  [pp,  220- 


5.  x+x-\-2x  = 2]  find  X. 

12.  Let  X denote  the  number  of  hours,  then  (30— 4):;^;  = 


208 

1,760 


; find  X. 


For  answers,  see  end  of  this  chapter,  page  106,  of  this 
Manual, 


Lesson  5 

Assign  the  even-numbered  problems  from  i4-2g,  giving 
any  explanations  that  are  deemed  necessary  by  taking  odd- 
numbered  problems. 

The  idea  here  is  a little  exercise  work  to  develop  mechan- 
ical skill  with  equations,  alternating  quickly  with  problem- 
work  at  once  using  the  skill  developed.  Do  not  crowd  all 
the  exercises  together,  and  then  all  the  problems.  Shifting 
the  attention  from  one  sort  of  work  to  the  other  affords  an 
element  of  variety,  that  will  result  in  more  and  better 
results,  than  concentration  on  all  of  first  one  thing  and  then 
of  the  other.  It  will  doubtless  be  thought  worth  while  by 
some  teachers  to  point  out  a number  of  the  different  types 
of  problem  and  show  the  pupil  that  all  of  a certain  type  can 
be  solved  by  a certain  typical  equation.  For  example, 
a type  form  for  motion  problems,  rowing  problems,  clock- 
problems,  mixture-problems,  etc.,  may  be  set  up,  and  then 
by  mere  substitution  any  problem  of  the  type  may  be  solved. 
This  sort  of  work  is  of  the  very  essence  of  mathematical 
thinking,  and  is  of  course  eminently  worth  while.  A prac- 
ticable plan  is  to  select  for  class-work  a problem  of  a certain 
type,  and  assign  for  home  work  others  like  it.  For  example, 
take  up  No.  5 as  class-work  and  assign  numbers  44,  45,  46, 
48,  49,  53,  as  home  work.  No.  21  may  illustrate  the  type 
of  22,  23,  and  24: 

No.  54  is  typical  of  55,  56,  and  57, 

No.  26  is  typical  of  27,  28,  29. 


Simple  Equation  in  One  Unknown 


103 


-223] 


No.  59  is  a type  of  such  as  60;  and  Nos.  4 and  6 are 
types  of  7,  8,  9,  ii,  12,  13,  15,  16,  17,  18,  19,  and  20. 

After  some  experience  on  the  above  plan,  have  pupils 
themselves  attempt  to  solve  and  apply  their  solutions  to 
type  problems,  entirely  unaided  by  the  teacher. 

See  end  of  this  chapter  for  answers. 

Notes  and  Suggestions  on  Problems 


15.  Let  X denote  the  actual  rate;  then  8(x+6)  — ii(x-“7) 
= 50.  Whence  = 25. 

16.  Let  X denote  the  rate  of  the  train  on  the  shorter  route, 
then  8^(0;— 10)  — 6a:=is.  x = /^o.  and  6:r  = 24o.  8^(x— 10) 
= 255- 


17.  Let  X denote  the  required  rate,  then 

80  20  100 

— hi 1= — • Then::t:  = 20. 

/v'  A/v*  'V 

vv  2*^  ^ 


18.  Let  X denote  the  required  rate,  then 

ty 

{h+t)x  = tr.  Whence  — . 

n-f-t 

20.  Let  X denote  the  number  of  miles,  then 

§+-  = 9.  Whence  x=i8. 
o 3 


X 

22.  The  equation  is  obviously,  lod — =x.  Whence 

12 

x = iohi-. 

X 

23.  The  equation  is  30+—=  0^.  Whence  0^  = 32 Also, 

I ^ I 

x+is  =— +15,  x=o. 


X 

24.  The  equation  is  x+30=3S+— . Whence 

26.  From  the  note  it  is  evident  that  = the  gain 

per  month,  and  (t\— A)  times  x (the  number  of  months 


104 


First-Year  Mathematics  Manual  [pp.  22 j- 


required  for  Venus  to  gain  a complete  revolution)  equals 
one  revolution.  Whence  x = 2o, 

27.  From  the  preceding  explanation  it  is  evident  that  the 

equation  is:  Whence  x = 5. 

28.  Similarly,  — ^x=i.  Whence  a;  = 2 9. 544. 

29.  Similarly,  Whence  ^ = • 

Lesson  6 

About  20  problems  selected  from  the  list  from 

Follow  the  plan  of  explaining  any  difficulties  thrown 
up  as  the  work  progresses,  by  solving  not  the  particular 
problem  in  which  the  difficulty  arose,  but  a similar  prob- 
lem. The  class-work  should  avoid  instilling  mere  imitative 
habits.  This  practice  should  be  abandoned  only  under 
exceptional  circumstances. 

See  end  of  this  chapter  for  answers. 

Notes  and  Suggestions  on  Problems 

45-51.  All  solved  by  the  second  method  explained  under 
Problem  44  of  the  text. 

52.  Let  X denote  the  number  of  ounces  added.  Then, 

^ . Whence,  x = 60. 

53.  Let  X denote  the  number  of  ounces  of  iron.  Then, 

^x+oc-{-^oc  = i2^.  Then,  x = ^2. 

54  — 57.  See  Problem  52. 

59.  Let  X denote  the  number  of  pounds  of  lead,  then 

11^+11(159-^)  = 143-  Then, 
a;  = 45,  and  159— x=ii4. 

60.  See  59, 


Simple  Equation  in  One  Unknown 


105 


-230] 


64.  Note.  . 06  X 100  = number  of  weight  units  of  salt  = 

.08(100— x).  (i)  X = 25, 

(2)  . 13(100— x)  = .06X 100,  and  x = 53|^. 


65-66.  See  64. 

67.  Let  X denote  the  number  of  percent  of  water 
‘ added,  then  .o3J(ioo+x)=  .06X100,  and  x = 7i|. 

68.  Let  X denote  the  percent  of  water  added,  then 
.8o(ioo+x)=  .95X100,  and  x=i8|. 

69.  Let  X denote  the  required  percent,  then  75^  = 
.30X100,  and  x = ^o. 

70.  See  69. 

72.  Let  X denote  the  number  of  dozens  in  the  box,  then 
i5:r  = 3o(x— 5)— 30,  and  x=i2. 

73.  Let  X denote  the  number  of  days  he  was  idle,  then 
2(20— x)— :r  = 34.  Whence  x=2. 


74.  See  73. 

X X 

75.  Let  X denote  the  distance  required,  then 
Whence  x=i8o. 


76.  See  75. 


oc  oc  oc  oc 

77.  Let  X denote  the  income,  then  — |-tH — \—i-\-e  = x, 

abed 

abode 


Solve  f or  X , x ^ __  ___  ___  . 


Lesson  7 

Exercise  XXXIII  and  the  odd-numbered  parts  of  Exer- 
cise XXXIV  to  Exercise  20.  Give  some  attention  to  the 
type- problems  of  the  assignment. 


io6 


First-Year  Mathematics  Manual  [pp,  2ji  - 


Lesson  8 


Finish  with  the  even-numbered  parts  of  Exercise  XXXIV. 
It  is  also  a good  plan  to  distribute  these  exercises  along 
through  the  subsequent  lessons  of  chapters  xi  and  xii. 


Lesson  9 


The  Summary  and  any  specific  outstanding  difficulties  of 
the  chapter. 

Emphasize  with  the  class  the  ideas  that  are  set  forth 
in  the  Summary.  These  are  the  specific  notions  the  pupil 
is  expected  to  carry  away  from  the  study  of  the  chapter. 
It  is  recommended  that  the  pupil  attempt  to  remember  these 
notions. 

Finally,  it  is  believed  that  better  results  will  be  secured 
from  the  sort  of  work  and  the  assignments  of  time  sug- 
gested above  than  will  be  obtained  from  a more  liberal 
allotment  of  time  to  it.  First-year  high-school  pupils  can 
attain  only  a limited  degree  of  thoroughness,  no  matter 
how  ambitious  the  teacher  for  higher  ideals.  Moderate 
thoroughness  and  movement  forward  through  the  subject 
with  some  appreciable  celerity  is  more  effective  than  over- 
insistence upon  the  adult  standard  of  thoroughness.  Firm- 
ness of  grasp  of  fundamentals,  and  conviction  of  the  worth 
and  scope  of  the  subject  are  most  effective  here. 


Answers 


Page  215: 


2.  48  5.  40,  50 

3.  6.  45,  13s 

4.  5 7.  30,  60,  90 


8.  6,  18 


9.  14 


Simple  Equation  in  One  Unknown  107 


2ig: 

2. 

6^ 

14. 

I 

25- 

3 

4 

3- 

10 

IS-  - 

-4 

26.  - 

3l  9 

4. 

— 2 

16. 

7 

27. 

20 

S- 

4 ■ 

17- 

6i 

28. 

20 

6. 

5 

18. 

8 

29. 

10 

7- 

3 

19. 

2 

30. 

8 

8. 

-5 

20. 

6 

31- 

2 

9* 

— 2 

21. 

8 

32- 

5 

10. 

-3 

22. 

4 

33- 

4 

II. 

3 

23-  - 

-I 

34- 

ifl 

12. 

54 

24. 

3 

35-  - 

-2. 

13- 

13 

Pa^e  220;  § 157. 

I. 

23,  7 

2. 

Change  Problem  2 to  read  20  sq. 

units,  etc. 

Ans.  8, 

3 

3* 

16 

5-  8,  8, 

16 

7- 

4- 

48 

6.  24 

221; 

8. 

3,780 

9; 

js  feet 

10. 

1,200  ’ 

m 

T 

12.  hour,  or  i6y\  seconds 
I 

i,76o(w— w) 


(i) 

10 

(6) 

30 

(ii) 

4 

(2) 

6 

(7) 

4 

(12) 

— I 

(3) 

I 

(8) 

— 2 

(13) 

a 

(4) 

4i 

(9) 

6 

(14) 

2C 

(s) 

3 

(10) 

6 

io8 


Firstly  ear  Mathematics  Manual  [pp.  222- 


Page  222: 

15-  25 

16.  4th  line,  change  to  8^  hours.  240,  255 

17.  20 


c 

w+w 

20.  18 

22.  10^  past  2 


Page  22 y, 

23.  o and32T\,  past  3 

24.  5rr,  past; 


25- 

(1) 

I ““6a 

2 

(5)- 

I 

(10) 

(11) 

1+^: 

I 

(2) 

I 

^“f-a 

(6) 

(7) 

18 

6f 

(12) 

4 

T 

(3) 

I fa 

T 

(8) 

9I 

(13) 

c-f-a 

(4) 

± 

a-f"^ 

(9) 

iA 

8 7 

26. 

20  mo. 

27.  s mo. 

28.  29.544  da. 


Page  224: 


ah 

36. 

6,  8 

29.  ~ — 
h — a 

37- 

24,  26 

31.  2 times 

38. 

a—4 

a+4 

■ 

32.  9,  10 

4 

4 

-22q] 


Simple  Equation  in  One  U nknown 


log 


Page  22 §: 

42.  13,  IS 

43-  (i)  a-z,x->rz,x—a 


45- 


(6) 


etc. 

(7) 

etc. 

(8) 

, etc. 

(9) 

(s)  i2/a2 

(10) 

1142?,  3,657-7 

47- 

37? 

90, 15,  IS 

48. 

12, 

j 44 1 7 669 

, etc. 


, etc. 


6—az 
<i-\-z 
4 

(8)  etc. 

a-\-6z+i2 
6s 


, etc 


Page  226: 


49. 

am 

bm 

62. 

i,iiii. 

888| 

a-\-b-\-c  ^ 

a+^»+c  ’ 

63- 

3,750 

cm 

64. 

{a)  25 

a-\-b-\-c 

{b)  53H 

51- 

li 

65- 

sA 

52- 

60 

70  c'n. 

looa  — 

looj 

53- 

12^;,  42  i., 

00. 

b 

54- 

190 

67. 

717 

55- 

20 

68. 

i8f  pet 

' cent 

56. 

124 

69. 

40 

57- 

be— ad 

d 

70. 

72. 

56 

144 

59- 

45,  114 

73- 

2 

60. 

194,  126 

ab  — d 

61. 

400,  600 

74- 

b+c  ' 

\pp.  234- 


CHAPTER  XI 

LINEAR  EQUATIONS  CONTAINING  TWO  OR  MORE 
UNKNOWN  NUMBERS— GRAPHIC  SOLUTION 
OF  EQUATIONS  AND  PROBLEMS 

Squared  paper  and  a small  ruler  for  laboratory  work  in 
class  will  be  needed  for  this  chapter  also.  Observe  that  the 
graphical  work  is  to  show  the  meaning  of  the  solution  of 
equations,  and  for  this  reason  it  should  come  before  the 
algebraic  solutions.  After  the  algebraic  method  of  solving 
equations  has  been  learned  it  is  so  much  less  laborious  to 
solve  equations  by  it,  that  pupils  will  not  care  to  continue 
the  graphical  procedure,  nor  should  they  be  required  to  do 
so  save  to  clear  away  particular  difficulties.  Pupils,  however, 
readily  learn  and  remember  for  a time  the  mere  technique 
of  algebraic  solutions,  and  do  not.  concern  themselves  about 
comprehending  its  significance  and  its  limitations.  They 
soon  come  to  apply  this  technique  purely  mechanically  and 
irrationally.  The  antecedent  work  with  the  graph,  with 
frequent  subsequent  appeal  to  it,  when  matters  are  known 
by  the  teacher  not  to  be  sufficiently  well  comprehended  by 
pupils,  will  avoid  thoughtless  and  oftentimes  foolish  steps 
and  conclusions.  Algebra  will  thus  be  made*  to  appeal  to 
pupils  as  a subject  that  has  to  do  with  thinking  rather  than 
with  mere  manipulating.  Too  often  the  high-school  teacher 
deludes  himself  with  the  notion  that  pupils  really  understand 
what  they  can  only  manipulate.  The  only  way  to  make 
algebra  appeal  permanently  as  a subject  worth  studying 
is  to  have  it  well  understood.  For  these  and  many  other 
educational  reasons  graphs  should  be  used  in  teaching 


no 


Linear  Equations 


III 


-239] 


simultaneous  equations  and  their  use  should  precede  the 
technically  algebraic  work. 


Lesson  i.  Arts,  160-61 

Sketch  rapidly  the  solution  of  Problem  i,  and  point  out 
on  the  graph  the  meanings  of  the  lines,  and  the  solution  of 
the  problem.  Have  the  class  solve  similarly  during  the 
recitation  period  Problems  2 and  3,  page  235.  Assign  4 
and  5 for  home  work.  In  a similar  way  work  through  the 
solution  of  Problem  i,  Art.  161,  and  Problems  2 and  3, 
page  237,  and  assign  4 and  5 as  home  work.  Recall  the 
law  of  leverages. 

Lesson  2.  Arts,  162-63 

Go  over  the  home  work,  clear  away  the  difficulties 
revealed  by  it,  read  with  the  class  the  text  of  Art.  162,  see 
that  it  is  understood,  and  work  with  them  Exercise  4, 
page  238.  Assign  the  rest  for  home  work.  Remember 
the  graphical  work  is  not  so  much  for  its  own  sake  as  to  make 
the  meaning  of  solving  equations  understood.  Do  not  dilate 
in  detail  about  the  graphs,  but  show  how  and  wherein  the 
graphs  furnish  the  solutions. 

Lesson  3.  Arts,  164-65-66 

After  clearing  away  the  difficulties  shown  by  the  pre- 
ceding day’s  home  work,  have  different  members  of  the 
class  read  and  tell  the  sense  of  the  text  of  Art.  164  and  of 
Art.  165  down  to  Exercise  XXXVI.  Solve  Exercises  i 
and  2 with  the  class  and  assign  the  rest  as  home  work. 
Make  the  meaning  of  equivalent  and  dependent  clear. 

Some  teachers  will  prefer  to  have  the  class  do  together 
the  problems  of  Arts.  164  and  165  as  a laboratory  exercise. 
This  is  also  a good  plan. 


1 1 2 First-  Y ear  M athematics  Manual  [ pp . 239  - 

Exericse  XXXVI 

Answers: 

j Equivalent  j Contradictory 

( Coincident  \ Parallel 

2.  Same  as  i.  5 and  6.  Same  as  4. 

3.  Same  as  i. 

Lesson  4.  Arts.  167-68-69 

After  5 or  10  minutes’  clearing  away  difficulties  of  pre- 
ceding lesson,  have  a pupil  read  the  problem  of  Art.  167 
and  show  that  Equations  (i)  and  (2)  express  the  condi- 
tions (a)  and  (b).  Then  have  a pupil  show  the  pertinency 
of  the  abbreviated  phrases  in  parenthesis  as  (Add.  Ax.). 
Have  another  pupil  read  and  answer  the  next  two  questions. 
Drop  the  rest  of  the  questions  around  rapidly  and  promiscu- 
ously in  the  class.  Keep  the  work  moving  briskly.  Con- 
tinue in  this  way  through  the  questions  of  the  next  Article. 

Then  work  Exercise  2 at  board,  and  have  Exercises 
I and  3 done  by  the  class  as  a laboratory  exercise.  Assign 
the  rest  of  Art.  169  as  home  work.  Insist  on  the  checking 


of  results. 

Exercise  XXXVII 

Answers: 

I-  I,  3 

M 

1 

M 

1 

7- 

L 

3 

2.  2,  3 

5-  I,  2 

8. 

2, 

I 

3-  12,  -3 

6.  3,4 

9- 

3, 

2. 

Lesson  5.  Arts.  170  and  171 

Work  through  Art.  170  carefully  as  was  suggested  above 
for  Art.  167,  solve  Exercise  4 on  board,  and  then  have  half 
a dozen  done  by  class  as  a laboratory  exercise.  Notice 
the  text  asks  to  have  the  class  go  through  the  list  of  exercises, 
first  telling  how  to  solve  each  one,  and  then  calls  for  the 
solutions.  This  practice  in  looking  at  equations  and  judg- 


-244] 


Linear  Equations 


113 


ing  what  to  do  before  beginning  the  actual  work  eliminates 
much  of  the  puttering  that  pupils  are  too  wont  to  practice 
when  they  do  not  know  either  how  to  start  or  to  proceed. 
Pupils  should  form  the  habit  of  deciding  first  what  is  to  be 
done,  what  is  given  to  do  it  with,  and  what  is  the  most 
economical  way  to  start.  Even  the  exercise  work,  properly 
administered,  can  be  made  to  contribute  to  this  habit. 
Assign  the  rest  of  the  list  of  exercises  to  be  done  as  home 
work. 

Exercise  XXXVIII 

Answers: 


I. 

3,  -2 

5- 

2,  I 

9- 

1, 2 

2. 

12,  3 

6. 

I,  2 

10. 

7,  II 

3- 

2,  I 

7- 

II,  10 

II. 

2, 1 

4- 

3,  2 

8. 

3,  2 

12. 

4,  8. 

Lesson  6.  Arts.  172-73 

Have  pupils  study  carefully  and  then  tell  how  to  eliminate 
by  addition  or  subtraction  and  how  by  comparison,  and 
then  compare  the  two  methods.  Let  them  attempt  to  tell 
when  they  would  use  the  one  or  the  other  method.  Do  what 
telling  you  deem  necessary  by  exhibiting  an  exercise  that  is 
conveniently  treated  by  one  method,  but  clumsy  by  the 
other.  Studying  consists  in  comparing  and  relating  things, 
and  the  way  to  study  methods  of  elimination  is  by  compar- 
ing, not  merely  committing  them.  Have  three  of  the  parts 
of  Exercise  XXXIX  worked  in  class  and  assign  the  rest  as 
home  work. 

Exercise  XXXIX 

Answers: 

1.  I,  2 

2.  6,  10 

3-  2,3 


4-  3,4 

5-  6,5 
6.  4,  5 


7.  8,  5 

8.  28,  10. 


1 14  First-Year  Mathematics  Manual  [pp.  244- 

Lesson  7.  Arts.  174-75 

This  lesson  should  be  reconnoitered  in  class,  and  4 of 
the  exercises  of  page  245  assigned  as  home  work. 

Exercise  XL 

Answers: 

1.  40,  45  3.  6s,  66  5.  22,  6 

2.  35,  24  4.  105,  63  6.  36,  60. 

Lesson  8.  Art.  176 

Treat  this  lesson  similarly  to  the  mode  suggested  for 
Lesson  7. 

Exercise  XLI 

Answers: 

Exercise  41. 


b a 

8. 

cie  bd  bd  ae 

a+6  ’ a-\-b 
b a 

ce—bf  ’ cd—af 
m^  — n^ 

cb+ci'^  ’ 

9- 

mk—nh  ^nk—mh 

ah  — bk  ak  — bh 

10. 

2(a"+fi")  3(a"+6") 

a^-b^  ’ a^-b^ 

a-\-b  ’ a—b 

/ 2c^d-jrd^~cd 

II. 

c,  d 

\ c^-\-bd 

12. 

(i-\~bj  d — b 

I 2bcd-\-cd—c^ 

13- 

h-\~kj  h — k 

\ c^A-bd 

14. 

2b— c,  2J+C 

b,  a 

d b 

Cib2 — C2bi  ^2^1  ““  ^1^2 

IS- 

d — b ^ d-\~b 

(Z162  ^^2^1  (llb2  djbl 

2kn  2kn 

7* 


-252] 


Linear  Equations 


IIS 


Lesson  9.  Arts.  177-78 
No  directions  are  needed  here.  Treat  as  above. 
Exercise  42. 

6,  5.  4 9-  5,  10,  20 

IS,  10,  S 

4,  S,  6 

5,  10,  20 


I-  I,  2,  3 
2.  I,  2,  3 
3-  I,  2,  3 
4.  10,  2,  I 


10.  I,  2,  3 

11.  3,  I,  2. 


Lesson  10.  Art.  179 

Work  half  a dozen  , of  these  problems  as  a laboratory 
exercise  in  the  class,  and  assign  half  a dozen  for  home  work. 

§ 179.  Answers 


1.  5c.  and  2jc. 

2.  $7.50  and  $4.75 

3.  72^,72^,36° 

4.  63^,27° 

5.  100°,  80°,  100°,  80° 


7-  25,  IS 

8.  5,  io  and  3,  5 

9.  40  mi.,  30  mi. 
10.  8,  3 

II*  39,20 

12.  $2,000,  $1,200 


6.  52!°,  127^° 

Lesson  ii.  Art.  179  continued  to  Exercise  XLIII 
Proceed  as  with  Lesson  io. 

§ 179 — Continued 


13.  $10,000,  $12,000 

14.  $10,000,  $12,000 


IS- 

16. 

17- 


12  in.,  10  in. 

Radii,  7,  21 
Areas  154,  1,386 


j Radii, 

\ Cirs.,  22, 


II 


18.  4,  10 

19.  10,  6, 


tr 


I 


I 


1 1 6 First-  Y ear  M athematics  Manual  [ pp . 252- 

Lesson  12.  Exercise  XLIII  to  Art,  180 

Take  one  laboratory  period  and  one  assignment  of 
home  work  on  the  work  to  Art.  180. 


Answers 


20. 


(1)  h I 

(2)  6,  12 

(3)  R,o 

(4)  12,  40 

(s)  14 

(6)  5,  7 

(7) 


21. 

¥,  f 

26. 

9,  -10 

22. 

4.8, 6.4 

28. 

I,  2 

23- 

1,  4 each 

29. 

2a/b 

24. 

10,  6 

30. 

6,  12  da. 

2S- 

iy  i 

31- 

'2— 

977  27. 

27. 

75 

-254] 


Linear  Equations 


117 


Notes  on  the  Problems  of  Two  and  Three  Unknowns 
P^e  253: 


m 


m — - 


ibn 


21. 


ni+n  = s,  “+«=S>  «=l.  »w  = ¥- 


£F=l/6^+8^  = io.  x = A’  ? = 

h ^.o  o 


( m = 

= 6.4.  /?  = !/ 6.4  • 3.6 

=4.8 

23-  (i)  1 

i ^x-3y=3 

[ %x—4y—iox--6y-\-3 

urjiM 

II 

y=4 

(2)  1 

\ 2x—y  = i 

1 2x—y=i 

x=^. 

y=4 

(3)  1 

j 20:— 3;+2=3 
j ioa;-6>'+3  = 4 

II 

II 

10 


Page  254: 

24.  2^^-{ii2+^x+3y)=2,x-2y+go 

j ^x+y  = ^6 

\ gx+4y  — 2=  112,  x=io.  y = 6 

25.  The  base  of  the  second  triangle  should  be  sx—2y—i. 

I CX~i”iav~2I 

The  equations  reduce  to  j >'  = -V" 


26. 


- = ~ = — /.  4x+33;  = 6 
10  7i  10  ^ 

6x+43;+2  12  16 

— = ^ = 9-  >-  = 


— 10. 


ii8 


First-Year  Mathematics  Manual  [pp.  255 


Lesson  13.  Art.  180 


A laboratory  period  during  which  eight  or  ten  exercises 
are  solved  by  the  class  together,  and  an  assignment  of  about 
ten  selected  exercises — not  necessarily  the  easiest — is 
enough  time  for  this  section.  The  exercises  selected  for  the 
laboratory  work  may  well  be  types  of  the  exercises  assigned 
for  home  work. 


Exercise  XLIV 


Answers: 


I. 

1 

1 

2. 

7,  15 

3- 

30,  10 

4- 

12,  6 

S- 

0,  -2 

6. 

j 2a-2,b-4c 

\ 2(1— 36+4C 

( (i-\-b — c 

7- 

{ a — b-\-c 

8. 

S,  6 

9- 

12,  4,  30 

10.  4,  S 

a^  — b^ 

a^-b^ 

12.  m+iz,  m — n 

13.  10,  8 

14.  I,  2 
IS-  3,4 


16. 


17- 


18. 


!_s  {n—i)d 
n 2 
^_2S—an 
n 

!_  2s—nl 
n 

2{ln—s) 

n{n—\) 

a = 6,  1 = 20 
d = \,  1 = 0 
= n = 2() 

(^  = 3^  ^ = 384 
f = 2,  n = S. 


-26 1]  Linear  Equations  119 

Lesson  14.  Art,  181 

Devote  one  laboratory  period  to  four  to  six  of  these  prob- 
lems and  exercises,  and  assign  for  home  work  ten  to  twelve 
of  them. 


Answers 

to  Problems,  § 

M 

00 

M 

10,  6,  4 

/ (i)  5, 

2,  I 

) (2)  I, 

2,  3 

) (3)  I, 

2,  5 

' U)  I, 

2,  3 

i (i)  6 

, 3,  3 

1 (2)  3: 

, 12,  21 

763 

io5>  52^ 

, 26i 

3.  2,  I. 

From  (i) 

= 10. 

x = 3 

From  (2) 

53'+3z^ 

= 13- 

y = 2 

From  (3) 

4^+32^ 

= 15- 

Z = 1 

72°,  80°, 

28°.  A+B+C  = 

180. 

A = 72 

iA+|B  = C.  B = 8o 
jA+yVB  = JC  + 30.  C = 28 

8.  $6,000,  $4,000,  $8,000. 

Lesson  15.  The  Summary 

Clear  away  any  outstanding  difficulties  of  the  chapter, 
and  have  members  of  the  class  read  and  illustrate  the  state- 
ments of  the  summary.  Impress  the  class  that  the  summary 
contains  the  substance  of  what  pupils  will  be  expected  here- 
after to  know  about  this  chapter. 


[pp.  262- 


CHAPTER  XII 
FRACTIONS 

In  the  work  already  covered  fractions  have  been  handled 
without  comment.  It  is  now  desirable  that  there  be  an 
explicit  treatment  of  algebraic  fractions,  such  as  to  show 
how  the  methods  derived  in  arithmetic  for  addition,  sub- 
traction, multiplication,  and  division  of  fractions  hold  with- 
out exception  in  algebra.  The  material  of  this  chapter  is 
designed  to  present  this,  and  as  the  text  is  full  but  little 
comment  is  needed. 

It  will  be  observed  how  the  plan  is  to  apply  each  prin- 
ciple to  algebraic  expressions  as  an  induction  from  its  use 
on  particular  numerical  cases.  Thus,  in  the  reduction  of 
fractions  the  first  seven  examples  deal  with  definite  numbers 
such  as  after  these  the  transition  to  abihx  is  easy. 

In  discussing  a fraction  such  as, 

particular  numerical  values  may  be  assigned  at  first  to  n\ 
thus  for  n = 7,  we  have 

x^^  x^ ' 

After  it  is  clear  that  in  each  case  we  get  i/x^  as  the  simple 
fraction,  it  is  well  to  recur  to  the  meaning  of  x^]  thus 

x*^  __{x  • X • X . . . . to  n factors) 

(x  • X • X . . . . to  n factors)  • x • x • x' 

The  x^s  cancel  in  pairs  except  for  the  last  three  in  the  denom- 
inator, so  that  the  result  is  i/xK 


120 


-274\ 


Fractions 


I2I 


It  is  well  to  point  out  that  as  in  reductions  with  numer- 
ical exponents  we  subtract  the  less  from  the  larger,  so  it  is 
best  to  do  the  same  with  mixed  exponents.  Thus: 

I I I 

as  for  , we  get  - , so  for  ^ we  get  = - . 

In  examples  such  as  i6,  Exercise  XLV,  we  have  cases 
where  form  is  important.  Pupils  need  but  a little  encourage- 
ment to  learn  that  the  parenthesis  may  be  regarded  as  a 
single  quantity  or  letter  and  handled  as  such. 

The  teacher  is  fortunate  who  by  this  time  has  overcome 
in  all  pupils  the  tendency  to  “cancel”  the  in  an  expres- 
sion such  as 

a 

a-\-b 

This  fallacy  will  doubtless  reappear  now.  A good  way  to 
impress  the  incorrectness  of  such  an  operation  is  to  try  it 
out  with  particular  numbers.  For  example, 


for  f of  a dollar,  or  40  cents,  is  not  equal  to  | of  a dollar  or 
33i  cents. 

Pupils  should  be  encouraged  to  use  this  method  of  check- 
ing a doubtful  point  by  substituting  particular  numbers, 
preferably  primes. 

This  last  item  reminds  us  that  the  present  chapter  con- 
sists entirely  of  pure  mathematics  and  of  drill  work.  Con- 
sequently the  teacher  should  supply  the  concrete  applica- 
tions which  arouse  interest,  such  as  above  where  the  abstract 
number  relations  were  made  concrete  in  terms  of  parts  of  a 
dollar. 

Eight  or  ten  recitations  give  ample  time  to  cover  the 
ground.  Sections  182-91  might  form  the  first  half  of  the 
work,  while  §§  192-97  and  a review,  the  last  half.  The 


122 


First-Year  Mathematics  Manual  [pp,  263- 


teacher  should,  as  in  former  chapters,  prepare  the  class  for  an 
assignment  of  home  work  by  solving  a few  problems  of  each 
type  with  the  class  and  showing  clearly,  in  each  case,  the 
principles  on  which  the  processes  depend. 


Answers  to  Exercises 
Exercise  XLV 


I. 

1 

II. 

b 

18. 

2. 

4 

9 

am 

n 

4 

u 

6' 

7 

12. 

— 

19. 

4* 

f 

vy 

5- 

1 3 

1 8 

13- 

I 

x^  , 

20. 

6. 

f 

I 

7- 

7 7 

11  1 

14. 

x^ 

21. 

8. 

a 

X 

IS- 

yl 

x^ 

22. 

9* 

b 

16. 

a 

23- 

y 

b{x-\-y) 

x^ 

17- 

x—y 

10. 

— 



yl 

m^ 

c 

r 

I 

m^n 

(a-\-hyi})-c) 

{c+iy 

I 


{u-\-vY 

{r—sY 

a^¥ 

a¥{a^+b^Y 


U 


-6 


Exercise  XL VI 
a-\-b 


c 

a—b 

c 

m-Yn-\-k 


10. 


II. 


r-s-Yt—v 

y 

I 

a-\-b 


X 


-267] 


I. 


2. 


x—y 


3*  *2^ 

x—y 


Fractions 

Exercise  XLVII 
a—h—c 

6.  a 
x+y 
a 

- c-\-ax 


123 


9*  I 

X 

a 

3^+26 
x—y 


X 

10.  - 
a 


II. 


25 


Exercise  XLIX 


2. 

3* 

4. 

5- 


tV 

6. 

b-c 

10. 

a—b 

A 

be 

(2a+36)(3a+2i») 

9 

7- 

m — t 

II. 

— 2y^ 

gx 

mi 

X4  — y4 

J'-Z 

8. 

a 

12. 

C 

2>y 

ab+¥ 

{a-\-b)  {a-{-b-\-c) 

x—b 

-2h 

2^+26  — 15 

bx 

9- 

a^-b^ 

13- 

3«+3* 

Exercise  L 


1. 

2. 
3* 

4- 

5- 

6. 


n 

7 

II. 

19(2+46 

A 

/ • 

a 

35 

7 9 

T32 

8x=*=2I 

8. 

00 

12. 

/^ax 

a^—x^ 

24 

dx^cy 

13- 

4(2  ^ 2 1 

36 

9* 

dy 

3(2x+2a^ 

2(2=1= 

10. 

x{b^a) 

14. 

2a^x^ 

10 

ab 

x^—a^  * 

124 


First-Year  Mathematics  Manual  [pp,  26j- 


Exercise  LI 


^ 5/>+2g+r 

10 

16. 

^ 86+sc4-2ic/ 

17- 

Aa 

18. 

x—y 

4- 

ax 

19. 

ay+bx 

cxy 

20. 

. a—b 

6. 

ca 

21. 

x—2az 

tj  

12^6 

22. 

8.  0 

23- 

I 

x^y^ 

24. 

a—b 

10. 

a—i 

25- 

I 

2(X— l) 

26. 

z^-\-y 

2x{2X  — 2,y) 

27. 

b{zd+a) 

3^/(a+26) 

28. 

a-\-bx 

14. 

X 

29. 

x — ay 

IS-  ^ 

y 

30- 

17a 

3 

26% 

3 

__x 

4 

ay-\-hx 

x^y^ 

2(2a+&) 

3X 

5^-7g 

15^ 

I— 

a^ 

a^+b^+c^ 

abc 

2y—x 

yix-y) 

2x^-}- 1 Sxy  — 1 23^* —4x2; +9^2 
Sxy 

^b^+5ab—4a^+^ac—4bc 

ab 

xy+xz—4 

x^y^ 

-b^ 

a{a-~b) 

{a— by  —4ad 

4ab 
gx 

x^—i 


-27 1\ 


Fractions 


125 


Exercise  LII 
Results  evident 


Exercise  LIII 


X6 

2 1 

5- 

acx 

1.5 

32 

hdy 

15^ 

6. 

28b 

6ty 

7- 

Exercise  LIV 


I. 

5^ 

IS- 

91: 

2. 

9- 

9f 

¥ 

3* 

c 

10. 

gaz 

16. 

ab 

IOC 

a^—b^ 

4- 

a 

35^' 

gxy 

5- 

a 

II. 

64/ 

17- 

4 

6. 

bx 

a 

12. 

sy 

7x 

18. 

(a-6)(2;-fy) 

(0+6)  (x—y) 

d 

13* 

^b^x^ 

19. 

a^+6a^+iia+6 

7- 

2iaxyz 

loy 

6a^+ii^“"6 

0 

az 

14. 

{a+by 

20. 

x(a+6) 

8. 

cx 

a^+b^ 

3(x-y)  ’ 

Exercise  LV 
Results  evident 


126 


I.  — 


ahz^ 

xy 

2ixy 

4 

2 


First-Year  Mathematics  Manual 

Exercise  LVI 
6abz^ 


[pp.  272- 


5- 

6. 


5xy 

3d 

aib^{4x-sy) 


xyz 


8.  2^ 


10. 


^h^xy^ 

Ta^c^ 


ybx^ 

II.  x^ 

13-  2*5(a;-l)(«-*) 

14-  A- 


ERRATA  IN  THE  TEXT  OF  FIRST -YEAR 
MATHEMATICS 

On  p.  265,  § 188,  in  No.  6,  for  ^ read  ^ ; on  p.  268,  in 
No.  31,  for  = read  — . 


CHAPTER  XIII 


-27s] 


FACTORING,  QUADRATICS,  RADICALS 

In  this  chapter  we  take  up  a matter  of  great  importance 
in  the  successful  use  of  algebra.  The  student  who  cannot 
factor  is  seriously  handicapped  in  all  future  work.  It  is 
hardly  possible  to  lay  too  much  stress  on  the  attainment  of 
proficiency  in  factoring. 

The  material  here  presented  may  be  classified  as  follows: 

{a)  Eight  elementary  methods  of  factoring  algebraic 
expressions,  viz. : 

Extraction  of  monomial  factors. 

Factoring  by  grouping  terms. 

Trinomial  squares. 

The  difference  of  two  squares. 

Trinomials  of  the  form  x^-\-ax-\-b. 

Trinomials  of  the  form  ax^-\-bx-\-c. 

The  sum  of  two  cubes,  and  the  difference  of  two  cubes. 

The  remainder  theorem.’’ 

{b)  Application  of  these  methods  to  the  solution  of 
quadratic  and  higher  equations;  and  an  introduction  to 
radicals. 

{c)  Review  material  applying  factoring  in  the  four  funda- 
mental operations. 

The  teacher  should  take  up  each  new  method  at  the 
board  first;  it  is  too  much  to  expect  the  pupils  to  get  up 
the  theory  by  reading  the  text.  Further,  it  is  not  expected 
that  all  examples  be  worked.  The  chapter  is  made  full  to 
afford  ample  material  from  which  each  teacher  may  make 
an  individually  satisfactory  selection. 

§§198-203  take  up  the  topics:  ^^What  is  a factor?” 
and  how  to  extract  monomial  factors. 


127 


128 


First-Year  Mathematics  Manual  [pp.  275- 


Many  pupils  have  much  trouble  with  factoring  because 
they  lack  a clear  notion  of  what  they  are  trying  to  do.  The 
first  object,  then,  should  be  to  show  clearly  how  a factor  of 
an  expression  is  an  exact  divisor  thereof,  and  then  to  drill 
on  this  notion.  The  class  should  be  called  upon  for  numerical 
examples  first,  after  which  § 200  may  be  taken  up  orally; 
and  then,  when  the  examples  become  too  difficult  for  clear 
work,  the  class  may  be  sent  to  the  board,  each  pupil  to 
put  on  in  order  all  he  can  of  the  list.  This  affords  the 
teacher  an  opportunity  for  personal  work  with  the  more 
backward. 

After  practice  in  factoring,  the  idea  that  “a  factor  is 
an  exact  divisor”  should  be  recalled  and  drilled  upon.  This 
may  be  done  by  remultiplying  the  factors,  and  by  substitut- 
ing particular  numbers  as  shown  on  page  276.  This  sort 
of  testing  should  never  be  dropped.  Later  when  the  expres- 
sions are  complicated  enough  it  is  well  to  have  pupils  actually 
use  long  division  to  divide  an  alleged  factor  into  an  expres- 
sion. By  every  method  strive  to  have  them  realize  clearly 
that  a factor  is  an  exact  divisor. 

Exercises  LVIII  and  LIX  contain  review  material 
with  applications  of  factoring.  The  first  list  might  be 
assigned  for  home  work  after  the  first  lesson.  The  next  day, 
after  review  and  discussion.  Exercise  LIX  can  be  considered, 
as  many  examples  as  possible  worked  in  class  and  the  re- 
mainder assigned  for  home  work.  At  the  third  meeting  the 
next  method  can  be  presented. 

§§  204  to  208  are  devoted  to  factoring  polynomials  by 
grouping  the  terms.  After  the  material  in  § 204  is  presented 
Exercise  LX  may  be  started  by  having  single  pupils  work 
the  first  examples  at  the  board.  This  is  a place  where  remul- 
tiplication and  long  division  are  invaluable  to  check  the  work 
and  to  bring  out  clearly  the  nature  of  the  method.  After 
several  of  the  simpler  examples  have  been  worked  separately. 


-283] 


Factoring,  Quadratics,  Radicals 


129 


general  board  work  may  be  used  as  before.  For  home  work 
examples  ii  to  20  and  in  § 206,  i to  5 might  be  assigned. 

At  the  next  recitation  the  more  difficult  examples  24 
to  30  should  be  considered.  Here  is  an  opportunity  to 
bring  up  the  notion  that  in  factoring  it  is  the  form  of  the 
expression  that  is  of  importance.  Thus  in  24  the  binomial 
factor  (a+w)  can  be  treated  as  a monomial  factor,  and  in 
25  and  26  (x+y)  appears  as  an  exact  divisor.  To  consider 
27,  begin  with  particular  numbers,  thus:  x = ^,  m = j,  y = 3. 
Then  the  pupils  will  have  little  difficulty  in  factoring 

^S+7-|.^7j3  + ^5674-j7+3. 

By  several  such  cases  the  underlying  form  can  be  brought 
into  evidence,  and  then 

a^+^+a^b^+a^b^+b^-^y 

can  be  factored. 

In  §§  209-10  we  consider  trinomial  squares.  To  have 
clearly  in  mind  the  result  of  squaring  a binomial  the  twenty- 
one  examples  in  § 209  are  given.  The  writer  has  found  it 
expedient  to  send  the  class  to  the  board  and  have  each 
example  put  on  by  a pupil.  With  the  products  all  in  evi- 
dence the  rule  for  squaring  a binomial  results  almost  auto- 
matically and  then  can  be  reversed  for  factoring.  It  is 
important  in  all  factoring  that  the  pupil  be  able  to  describe 
the  procedure  in  words.  In  the  present  case  each  pupil  should 
be  led  to  state  substantially  this  method:  ^^Take  the  roots 
of  the  perfect  squares  and  see  if  their  product  is  half  the  other 
term.” 

In  Problems  20-26  we  again  have  illustration  of  the 
importance  of  recognizing  form.  Some  of  these  examples 
should  be  considered  in  class  first.  Thus  in  20  suggest  the 
substitution  A^{m-\-n).  The  factors  of  ^^+2^  + 1 are 
evident  at  once;  then  the  term  {m-\-n)  may  be  restored 
wherever  A appears. 


130  First-Year  Mathematics  Manual  {pp,  28 j- 

This  device  of  substitution  to  bring  out  the  form  of  the 
whole  expression  is  a powerful  instrument  throughout  all 
algebra,  and  the  students  should  be  encouraged  to  use  it 
freely. 

The  practical  application  in  Problem  29  of  the  squaring 
process  usually  arouses  interest.  The  examples  should  be 
worked  orally. 

§211  is  a review  exercise  which  might  be  assigned  for 
home  work  and  graded  with  the  value  of  a quiz. 

We  proceed  to  the  important  use  of  factoring  trinomial 
squares  in  the  solution  of  a quadratic  equation.  That 
“quadratic’’  means  second-degree  needs  emphasis.  The 
first  lesson  should  be  taken  up  with  a graphical  study  show- 
ing how  a quadratic  expression  takes  on  different  values  for 
different  values  of  the  unknown.  Not  only  should  the 
graph  on  page  284  be  reproduced  but  the  first  two  examples 
on  page  285  should  be  worked  in  class,  different  pupils  being 
called  upon  to  compute,  mentally,  the  expression-value  for 
a given  x and  then  others  required  to  come  to  the  board  and 
plot  the  points.  The  remaining  problems,  3-8,  should  be 
assigned  in  pairs  to  different  groups  in  the  class  for  home 
work.  At  the  next  recitation  the  graphs  can  be  copied  on 
the  board;  after  criticism  thereof  we  are  ready  to  bring  out 
the  idea  of  the  quadratic  equation  as  a condition  to  he  satisfied. 
To  every  quadratic  expression  there  corresponds  a quadratic 
equation  which  sets  the  condition  that  the  expression  be 
zero.  This  condition  is  satisfied  only  by  the  values  for  x 
at  the  points  where  the  graph  cuts  the  :r-axis. 

The  algebraic  method  of  solution  can  be  presented  now, 
Exercise  LXVI  being  taken  orally.  During  the  work  at 
the  board  it  is  advantageous  to  get  from  each  pupil  a descrip- 
tion of  his  method  for  finding  how  to  complete  the  square. 

The  solution  of  quadratic  equations  by  completing  the 
square  naturally  brings  to  light  equations  such  that  when 


“2p5]  Factorings  Quadratics , Radicals  13 1 

the  left  member  has  been  made  a perfect  square,  the  right 
member  is  not.  The  only  difficulty  is  to  give  to  radical 
numbers  a meaning  which  will  be  both  clear  to  the  students 
and  also  accurate  enough  for  work.  This  topic  is  considered 
in  §§223-30;  a fuller  study  of  radicals  comes  in  the  next 
chapter.  The  graphical  solution  aids  greatly  in  the  algebra 
here.  It  is  well  to  have  the  four  graphs  on  page  292  plotted 
at  home  and  ready  for  reference  when  the  topic  is  begun  in 
class.  Recall  that  when  we  found  no  integer  to  represent 
the  quotient  of  5-^-7  we  created  a new  kind  of  number  by 
representing  the  division  thus:  f.  In  like  manner,  when  we 
can  find  neither  integer  nor  fraction  such  that  its  square  is, 
say  7,  we  make  a new  number  by  indicating  the  process  thus, 
1/  7.*  Pupils  take  hold  of  radicals  readily. 

When  the  drill  problems  as  equations  have  been  solved 
a further  examination  of  radicals  may  be  begun.  In  pre- 
senting the  material  of  §§  225-28  it  is  well  to  do  on  the 
board  all  the  arithmetic  that  is  merely  indicated  there;  a 
previous  rehearsal  on  paper  will  insure  smooth  and  rapid 
progress  at  the  board. 

For  drill  in  the  use  of  all  this  information  actual  substitu- 
tion of  irrational  roots  in  the  equation  is  invaluable.  A 
convenient  form  for  the  work  is  as  follows: 


Prove  [page  297,  ii  (7)]  that 
equation 


-11+51/5 

2 


is  a root  of  the 


If 


q^-\-iiq—i  =0. 

,_-ii+Sl/5 
? ’ 


* It  is  of  interest  to  tell  the  class  how  historically  this  radical  sign 
l/  ~ is  a conventionalization  of  a script  T written  before  the  number  as 
an  abbreviation  for  radix  or  root,  when  a useful  purpose  thereof  is  evi- 
dent as  is  the  case  in  solving  the  equation  here. 


132 


First-Year  Mathematics  Manual  [pp.  2Qj- 


I2I 

- 1101/5+25(5)  _ 123-55V/5 

4 2 

ii?= 

_-i2i+55v+ 

2 

I 

— 2 

2 

adding^*+ii^— 1=  = o 

Such  substitutiou  not  only  drills  in  the  manipulation  of 
radical  expressions  but  also  arouses  a clearer  understanding 
of  what  a root  does  for  the  equation.  § 230  furnishes  drill 
in  the  arithmetic  operations  involved  in  this  substitution. 

In  Exercise  LXVIII,  the  factoring  of  the  difference  of 
two  squares  may  be  presented  in  the  same  manner  as  that 
detailed  for  the  trinomial  square  and  for  factoring  by  group- 
ing. In  Exercise  LXIX,  again  the  recognition  of  forms 
is  important.  In  considering  such  expressions  as 
Problem  20,  the  main  point  is  to  show  that  if  we  divide  the 
exponent  of  by  2 we  get  the  square  root  of  for 

qS  . ^5  = ^5+5  = ^10^ 

In  such  a problem  as  41,  viz.: 

QX*  + 1 6y^ — 49a^ — 46*  — 2 8a6 + 2^xy, 

it  is  sufficient  usually  to  recall  that  we  are  free  to  rearrange 
the  terms  so  as  to  have  two  perfect  squares  in  evidence;  thus 

(9X^+24:ry+i6y^)  — (49^*+ 28^6+46*). 

§§  237-41  may  be  conveniently  given  together  in  three 
lessons.  The  material  of  § 237  should  be  presented  with  the 
first  few  examples  in  evidence  on  the  board.  It  is  well  to 
call  attention  to  the  following  facts.  In  x^-\-bx+c,  if  c is 
positive  then  both  numbers  in  the  factors  are  of  like  signs: 
plus  if  b is  plus,  negative  if  b is  negative.  If  c is  negative 


-305] 


Factoring,  Quadratics,  Radicals 


133 


then  the  two  numbers  have  unlike  signs  and  the  larger  has 
the  same  sign  as  b. 

In  § 238  the  examples  21-30  are  rather  difficult  and  may 
be  omitted  with  a weak  class.  In  factoring  one  such  as 

(24)  - 2ok^  LT  - 

it  is  helpful  to  arrange  the  letters  in  the  factors  first.  Thus 
dividing  2in  and  2r^  by  2 we  have 

L^){k^  LT). 

Once  these  are  determined  the  arrangement  of  the  numerical 
coeflScients  is  not  difficult. 

In  applying  this  method  of  factoring  to  the  solution  of 
quadratic  equations  a new  principle  is  used  which  is  quite 
different  from  that  of  completing  the  square.  After  the  left 
member  is  factored  it  is  well,  first,  to  recall  that  a solution 
is  any  number  such  that  when  it  is  put  in  place  of  x the 
expression  reduces  to  zero.  Second,  that  if  any  number  be 
multiplied  by  zero  the  result  is  zero.  Finally,  we  remark 
that  in,  say, 

{x-^){x+6)=o 

while  di  T iox  X will  make  the  second  factor  13,  it  makes  the 
first  factor  zero,  and  0X13  = 0.  This  is  the  chief  difficulty. 
The  teacher  will  find  that  substitution  of  the  particular 
value  of  X in  both  factors  at  once  helps  to  clear  away  a vague 
notion  that  in  saying  x=7  or  x = — 6,  we  are  each  time  arbi- 
trarily ignoring  half  of  our  problem.  Here,  as  in  all  quadrat- 
ics, a check,  by  substitution  in  the  original  equation,  should 
be  required. 

By  reducing  a quadratic  equation  to  factors  it  becomes 
evident  how  the  equation  can  be  made  for  given  roots. 
Thus  §§242-43  follow  naturally.  Still  this  may  well  be 
omitted  with  a weak  class. 

Now  the  factorization  of  trinomials  of  the  form  ax^-\-bx-\-c 


134 


First-Year  Mathematics  Manual  [pp,  30^- 


follows  naturally.  The  material  of  the  text  may  be  pre- 
sented after  the  manner  already  suggested.  Individual  help 
to  students  at  the  board  is  most  useful  here.  Pupils  can 
understand  the  problems  when  worked  by  someone  else, 
but  are  backward  when  set  to  do  the  work  by  themselves. 
The  principal  cause  is  reluctance  to  try  out  all  possible 
combinations,  or  lack  of  system  in  so  doing.  The  correction 
of  all  this  requires  individual  encouragement. 

The  application  of  this  method  of  factoring  to  quadratic 
equations  follows  in  § 254. 

In  §§  256-57  factorization  of  the  sum  or  the  difference 
of  two  cubes  is  considered.  The  material  may  be  presented 
in  the  manner  suggested  for  the  trinomial  square.  To 
combat  the  tendency  to  write 

(a^  — b^)  = {a—b){a'^-\-2ab-Yb^), 

remultiplication  and  long  division  are  very  effective.  It  is 
helpful  to  remark:  that  the  binomial  factor  is  always  the 
cube  roots  of  the  quantities  in  the  factor  and  connected  by 
the  same  sign  as  these,  while  the  trinomial  factor  has  for 
sign  on  the  cross-product  term  the  opposite  to  the  middle 
sign  in  the  binomial  factor. 

The  text  deals  quite  fully  with  the  remainder  theorem 
in  §§  258-67.  All  the  examples  in  § 258  should  be  worked 
fully  on  the  board  at  once,  for  the  design  is  to  get  the  remain- 
der theorem  stated  as  an  induction.  By  the  time  a fairly 
evident  relation  has  appeared  in  some  twenty  cases,  only  a 
very  backward  class  will  fail  to  perceive  it.  Once  it  is 
perceived,  the  general  proof  (§  261)  follows  easily. 

§ 262  is  designed  to  bring  out  the  fact  that  the  constants 
in  all  exact  factors  of  a polynomial  are  exact  divisors  of  the 
last  term.  This  helps  us  to  tell  what  numbers  only  need 
be  tried  in  factoring  the  polynomial. 

The  following  schedule  of  recitations  is  suggested: 


Factoring,  Quadratics,  Radicals 


-323] 


13s 


§§  198-201 2 recitations 

204-206 2 

209-210 2 

212-222 3 

223-230 3 

233  2 

.237-241 3 

247-248,  253-255 2 

256-257 2 

258-267.. 3 


Total  24  recitations 

[242-243] I recitation 


It  will  be  remarked  that  very  slight  comment  or  none 
at  all  has  been  made  upon  certain  sections,  namely:  202, 
203,  207,  208,  211,  231,  232,  234-36,  244-46,  249-52. 
These  are  the  review  sections  classified  under  {c)  in  our 
opening  remarks.  The  teacher  will  use  these  as  the  time 
for  the  course  permits  and  as  individual  judgment  approves. 
In  the  schedule  above,  time  has  been  allowed  for  most  of 
these  sections.  The  same  is  to  be  said  of  the  miscellaneous 
exercises  (§§  268-74)  with  which  the  chapter  closes. 


Answers 

Exercise  LVII 

Results  evident 

Exercise  LVIII 

x+y 

4xy 

2(a-j-b) 

5- 

yz 

4- 

6 

3C+4a6 

136 


First-Year  Mathematics  Manual  [pp.  277- 


7- 

3/^3 

10. 

bc+2ac—:^ab 

abcix-^-y) 

8. 

tX'f 

II. 

2 

3 

22 

14.  i 

9- 

I 

12. 

3(2:-3) 

5^' 

Exercise  LIX 

2. 

6. 

x = d 

9.  y = c 

3- 

y = 2m 

7- 

m = 6w^ 

10.  k = ab 

4- 

10 

11 

Ei 

8. 

d 

a 

S- 

k=m 

^ a-\-b—c 

II.  0;  = — r—  . 

w+z; 

§203 

Sm 

5 

Sk  21 

be  ac 

I. 

m+1  ’ 

m+i 

2 

1^ 

1 

1 

1 

m{u+v) 

4- 

m—n 

, a. 

, ar,  ars. 

Exercise  LX 

I. 

{a+b){x+m) 

16. 

2. 

la+b)ir+s) 

17- 

3- 

(a+&)  {d-{-t) 

18. 

4- 

{a+b){s+y) 

19. 

5- 

(cl — 6)(^”|-0 

20. 

6. 

(a—b)(x^-]ry^ 

21. 

7- 

(db  ~\~'Li)(c-]r  0^ 

22. 

8. 

(a^-\-b‘^) 

23- 

9- 

(5a+w)  (u—v) 

24. 

10. 

(a+m)  (i+ma)ma 

25- 

II. 

(cL’-\-b^  (cl — d) 

26. 

12. 

(x3+i)(x^+S)x 

27. 

13- 

(2:^-3)  (3^- S>') 

28. 

14. 

(w"+3)(2w+i) 

29. 

15- 

(^^+^)(3^-S) 

30- 

(i+3»'0(3-S>')3 

(2g+3a)(4^+5*) 

(S2-2)(3-4w) 

{b-\-x){2,a-\-2x-\-i) 

{i-\-r){x—r^xy) 

(i— x)(a;^+i) 
{a-\-m){c—n) 
{x-\-y){a-c) 

{x-\-y)  (i  -\-mx-\-my) 

(2:v:’'— 3/)(*— 3’) 

(3  w* — 5 w^)  (w“+»*) 


-28 j]  Factoring,  Quadratics,  Radicals  137 


Exercise  LXI 

x+m 

2:3+7 

7- 

r+s 

4- 

20:^+ 6 

3+« 

5- 

0-55 

8. 

Sb+2k 

50-5 

{b+c)  {m-{-k){a 

a—b 

m+n 

6. 

1 ^+3’ 

9* 

6y 

1.  x = c-]rd 

cd 

2.  x=^-^ 

^ah 


Exercise  LXII 


6.  y = 


3.  x= 


ah-\-ac-\-hc 
a—b 
n — k 


4.  w=- 

z 

5.  V = S + 2W, 

V—K 

w = — - 


ad+ac—bc 


a—b 


8.  m=- 


7.  x = d-\-c 
4cd 
'sab 
9.  u=^ 
10.  x=7 
II-  3’=ii- 


I.  i+O)  r{i-\-a) 


§ 208 


2.  a+6;  20+25. 


Exercise  LXIII 
Results  evident 


1.  (2:+y)’ 

2.  (a— by 

3.  (w+2w)* 

4.  (20—1)* 

5-  (*-3)' 

6.  (5+4)’ 

7.  (0+55)" 


Exercise  LXIV 

8.  {2m—2,ay 

9-  (S+S'-)^ 

10.  (iio+9y)“ 

11.  (c-8)» 

12.  (^+15)^ 

13.  (7  — low*)* 

14.  (a5c+4)* 


First-Year  Mathematics  Manual  [pp.  28^- 


IS- 

-113^)^ 

21.  {u-\-V-\-2tY 

16. 

22.  (3^+r+s)^ 

17- 

{^k-\-gywY 

23.  {a-\-h-\-c-\-dy 

18. 

(13^ 

-2,py 

24.  (w+w+i)* 

19. 

(iS-s 

®-i4cS)^ 

25.  {m-\-n-\-zo)^ 

20. 

{m-\-n-Y'^Y 

26. 

29. 

(i) 

169 

(7) 

361  (13) 

00 

M 

(2) 

196 

(8) 

324  (14) 

7,921 

(3) 

225 

(9) 

1,849  (15) 

4,489 

(4) 

441 

(10) 

1,444  (16) 

10,609. 

(5) 

484 

(ii) 

5,184 

(6) 

961 

(12) 

6,561 

Exercise  LXV 


I.  k^{a  — bY 

6. 

(a;+i)(s*'i+4a;^+3) 

2.  xy{x^+y'^){x’ 

-y)  7- 

{c-\-dy 

3.  a(2:^+i)(a;+i)  8. 

4.  76(a:+33')" 

9- 

{m-yn)  (3a+2)^ 

5.  2,xy{2a-yiy 

10. 

{x-\-y){x—y)2y. 

Exercise  LXVI 

Results  evident 

§ 221 

2-  i,  -4 

3- 

h -1 

4-  (i)  1,  ~§ 

(3)  1,-4 

(s)  4,  — f 

(2)  1,-1 

(4)  f,  -12 

(6) 

5-  (i)  -3 

(2)  -I,  -3 

(3)  I.  -5 

(4)  2,  -10 

(5)  -5.  -9 


(6)  3.  -17 

(7)  -5.  17 

(8)  —2,  12 

(9)  -7, 13 

(10)  -2,  -I. 


Factoring,  Quadratics,  Radicals 


139 


-297] 

(11)  -2,  -5 

(12)  5,  8 

(13)  6,  -7 

(14)  —m,  -sm 

1.  6,  10  ; 

2.  6,  8 ^ 

2.  (i)  jw=-3±|/s 

(2)  -3  =^1/3 

3-  (i)  -5=^V5 

(2)  -5=^l/3 

(3)  — 6=^i/3 

(4)  -6±4/7_ 

(5)  7=^  21/2 

7.  32+131/2 

8.  51+61/6 
87-191/3 

II.  (i)  r = ^ 

2 

(,)  j=^2ivrl2 

2 

(3)  X=-2±l/S 

(4)  y=3=^i/ii 


(15)  2,  6 

(16)  2,  f 

(17)  -3.  5 

(18)  — -f,  — 

§ 222 

• 5.6  S.  22 

• 4,  10 

§ 224 


(3) 

-3,  * 

-3 

(4) 

-3=*= 

i/“i 

(6) 

_7=t 

l/'7 

(7) 

1/5 

(8) 

-8± 

1/2 

(9) 

3=^ 

l/5 

(10) 

4=^ 

VII. 

120— 23I/  8 
50+131/10 

(5)  w=5±i/i9 

(6) 

2 

(7)  ,==ii+!^= 

(8)  5 = — io=t  8|/  2. 


, m 
^ 3a^+2b^ 


Exercise  LXVII 


3* 


u 

ski 

x+i 


4- 


140 


First  Year  Mathematics  Manual  [pp,  2gy- 


3(x+2> 


7.  — — 

I0(2X  — 3)* 

8.  2t+l 

9.  (4a -3&)  (2a -96). 


8^—101 


Exercise  LXVIII 
Results  evident 


Exercise  LXIX 


44.  {a^+b^+2>0'V){a^+h^—2>(^b) 

45*  (a;^+x+i)(:!i£;^— ^ic+i) 

46.  {^x^—y^+2>^y){/[x^—y^—2>^y) 

47*  (5^"+4}'"+3^3')(S^"+4/“3^3') 

48.  + 1 ) (x4 — + 1 ) 

49.  {Ta^b^—2x^—s^bx)  {Ta^b^—2x^+^abx) 

SI.  (i)  399  (6)  891 

(2)  396  (7)  884 

(3)  391  (8)  1.599 

(4)  899  (9)  1,596 

(5)  896  (10)  4,891 


(11)  8,096 

(12)  4,875 

(13)  8,091 

(14)  8,075 

(15)  9,996. 


Exercise  LXX 


m{x+y) 

a-\-b 


4a—7^c—d+2b 
^a—7,c+d—2b 
^ a+2b+2,c 
a+2i— 3^; 


I 


{x—a){x—b) 


k^{m^—mn-]rn^) 

a^ 


^ i-\-a-\-b 


2(m^+4) 

6^4—56 


4. 


Factorings  Quadratics,  Radicals 


-302] 


141 


§ 235 

(a+4)(<^+8) 

(<i-4)((Z-8) 

2 {2X-3y){x-\-a->rh) 

a(2x+33;)(x— a— &) 
2ah{a—h){2a— 

2a+sb^ 

ab{mx+n) 

x{i+ab){2x+sy) 


(a^+b^){r+s) 

{a^-b^){c+b) 


2X  — 3 

r^—1  * 


Exercise  LXXI 


1.  k = a—b 

2.  x=a+3& 

3.  w=is  or  -5 

n 

4.  m— z,n  = m{a--b) 

a—b 

5.  r=ii, -4 

6.  /=3A3 


7.  x — m-n 
a— 2& 


8.  ii£:  = 


a-\-2b 


9- 


10. 


m = i2,  —4 

g = ^—y  t = g{u  — v). 
u—v 


Exercise  LXXII 


1.  {x+2){x+l) 

2.  (w+2)(jM+3) 

3.  (k+5b){k+7b) 

4-  (3'+5)(>'+9) 

5-  (^-5)(^-8) 

6.  (a— 2)(a+io) 

7.  (g-i2)(g+is) 

8-  (&+3c)(6+i6c) 
9.  (g3-6)(?5+7) 

10.  (i+3)0-i7) 

11.  (r+i75)(r— 19^) 


12.  (v-\-7w)(v—i^w) 

13.  (a6+iic^)(a6— ISC*) 

14.  (z*+2)(3*— 12) 

IS-  (3'+S)(3'-i7) 

16.  (J+3)(J-io) 

17-  iP+9)(p-^°) 

18.  (»-+3)(r-i6) 

19-  {k+5b)ik+37b) 

20.  (x3;+7z)(a;3’— 16) 

21.  (r“— 3)(r“— 16) 

22. 


142 


First-Year  Mathematics  Manual  [pp.  jo2~ 


23.  27.  (a+6  — 2)(a+6— 5) 

24.  {k^-\-2>Lr){k”'—2iLr)  28.  (a+6-2)(a+6+s) 

25.  {fs9-\-Z‘D‘w){rsP  — 2iifw)  29.  (a— 36— 4(;)(a— 36+iic) 

26.  (»j+»+2)(w+w+3)  30.  x{m‘-\-26){m‘—2$). 


Exercise  LXXIII 


I. 

I,  2 

8. 

3, 17 

IS- 

5.  7 

2. 

I.  3 

9- 

-7,  II 

16. 

1 

M 

0 

3- 

-I,  2 

10. 

7, 16 

17- 

7,  -13 

4- 

I,  -6 

II. 

-2,  5 

18. 

2,  5 

5- 

-I,  -6 

12. 

-2,  15 

19. 

2,  5 

6. 

— 2,  6 

13- 

-2,  14 

20. 

-I,  8. 

7- 

7,  -8 

14. 

-6,  9 

Exercise  LXXIV 

I. 

10,  6 

3.  10, 20 

S-  3.6. 

2. 

0 

00 

o' 

4.  250 

§242.  3 

I. 

x^  — 6x-]r^ 

6. 

2x*+i3x+6  = o 

2. 

x^+2>^-"io 

7- 

0 

11 

<N 

1 

\ 

3* 

X^  — 2X—l^ 

8. 

X^  + 2X  — 63=0 

4. 

x^+Sx+iS 

9- 

.T^+7x— 60  = 0 

5- 

9X+4  = o 

10. 

§ 243-  3 

— {a-Yb)x+ab  = 

I. 

x^  — 4X+i=o 

6. 

A;*+6n;+6  = o 

2. 

X*  — 6xH-6  = o 

7- 

x*+6:r+4  = o 

3- 

x^--6x+4  = o 

8. 

x'^-\-Sx-\-g  = o 

4- 

X*  — 8:x:+9  = o 

9- 

:r^+io:x:+i8  = o 

5* 

jc^--iox+iS  = o 

10. 

X^+I2X+IS=0. 

-jod]  Factoring,  Quadratics,  Radicals 

Exercise  LXXV 

1.  ic^(a+2)(a— 4) 

2.  (w+w)(6+3)(6-is) 

3.  (/>-i)(/»+i)(/>-s)(/>+s) 

4.  (/>-2)(/>  + 2)(/)^+4) 

5.  2{a-\-b)(x-\-y -z) 

6.  (d''+w‘')  (D+ze<)  {v—w) 

7.  (jw— w+c+(i)(w— w— c— ^^) 

8.  (2a;“-7)(4a;-3) 

9.  2>i^—l){k—l-\-2m) 

10.  (2:-4y)(i-32) 

11.  [(w+«)— 7(c+«^)][(m+w)— 4(c+^01 

12.  flte(c+s)(c— 15) 

13.  ioc^{()a^-\-b‘){2,<i+h){ia-b) 

14.  (a“+4&3“)(a“-963“) 

15.  {m‘—2mn-\-2n‘){m^-\-2nm-{-2n^). 


X—2 
I.  

x-2, 

x-trb 

2»  j ’ 

ic+2a 


§245 

2a+3^ 

2+6 

a+b 


m—S 

^+6 


6. 


§ 246 


j (g+5)(2g+3)  4- 

^ — g 

2 4 2;-y+2' 

(i— m)(m— 3) 

11—3;^ 

(y_2)(y-5)(),-7) 


143 


3- 


Cn  IC-K) 


144 


First-Year  Mathematics  Manual  [pp.  308- 


Exercise  LXXVI 
§ 248 


1.  (2X-Y3)ix+4) 

2.  (c“f-6)(8c — 2) 

3-  (^-5)(3^-2) 

4-  (z-3)(8z-7) 

5-  (x-7)i5x-3) 

6.  (a—:2b){iia—b) 

7.  (^+i8)(7^-3) 

8.  («+3s)(i2<-5s) 

9.  (5?M— 9w)(w— 4«) 
10.  (2r-5)(5r+i) 


11.  (3i-7)(2&-5) 

12.  (3/-ii)(2/+7) 

13.  (6-a)(i7+a) 

14-  (S-z)(3+8z) 

15-  {i-8xy){i-xy) 

16.  (a;"-|-4)(22;“+3) 

17.  (2:j:“+7y“)(7:»;“+2>'“) 

18.  (2a+26-3)(3o+3J+i) 

19.  {x-\-y-\-z){Ax+4y-\-^^) 

20.  (c-</-fi)(3c-3J-s). 


8 249 

1.  xy{c—6){2c—i) 

2.  (32;+2)(cK^-3) 

3.  (x+i)*(x-i)= 

4.  6a(a—2b){a—^b) 

5.  ((i+6+c)(o-6-c+i) 

6.  (a— J)(a-^-&-f-i) 

7.  (4a:"+9)(2x+3)(22:-3) 

8.  o(ff''-l-i6)(fl^+4)(a+2)(a— 2) 

9.  (2(1— i)(a— i)(2o^+3<i—i) 

10.  (a— ii)(c+7)6^ 

11.  3(»jiH-2)(»i’H- r c I 

12.  (6“— 2i)(6°— 3) 

13-  (<1-5)" 


IS-  7(''-  i)(^+-i) 
16.  (a^-\-b^){t^-\-r-s). 


Factoring,  Quadratics,  Radicals 
Exercise  LXXVII 


-3^o\ 


I4S 


W + 2 

4* 

3(/+2 

zCy-s) 

w+3 

4<^+5 

'■  a{2y-7) 

2X+3 

5- 

o(w+4) 

0 22:-36 

3^+4 

2W+3 

2<I+3 

3a+5 

6. 

5^+2 

3^+3 

3(3 -4^>) 

§ 251 

. y{y+2){y-2)-\-{2y-i 

{2b-i){2b-3) 

4. 

(y+i)(2y-3)(3y-4) 

-(33^+8)  w{y- 12) 

(2c— i)(c+4)(9C+5)  y(w+i2)  ■ 

b{b+3x) 
x(b—sx) 


§ 252 

bn^+bn—2b—^m 

m{2m—$) 

2aw?  — ^am-\-'^m 
n^-\-n—2 


^ _ y(3g"+35) 

2a^+l7^^+2I 

_x{2a^-\-\^a-\-2i) 

3«"+35 


2 ^ 

i$x^-\-xy—6y 

2x^-x(7-l)-is 

x^-20 


w(2a^  + 2i) 

^^+5^+15 

OT(a;^+5^+i5) 

2x^+2 1 


3.  w=+|-x,  -^x 
x=^m,  —\m 

4.  k = ll,-^l 
l=-%k,  +P 


§ 254 

5-  -3.1- 

6.  m=^n,  \n 
n=%m, 

7.  c = ^dj  6d 

d=2C, 

8.  a = \b,  — 


146 


First-Year  Mathematics  Manual  [pp.  jio- 


9.  x=ly,^y,  y=^x,lx  13.  2,-3 

10.  U='^,  -\m  14.  3, 

II-  3,+i  IS-  0,1- 

12.  3,  -V- 


I. 


2. 


I. 


2. 

3- 

4- 

5- 
6. 

7- 

8. 

9- 

10. 


I. 


2. 

3- 


4- 

5- 
6. 


§255 

4,¥-  3-  4,  3f  S-  6, 

12,  9 4.  25,  40 

§ 263 

(o:— i)(^t:^+x+i) 

(a:-3)(2:-s)(a:+i) 

ix-s)(x-4)(x+2) 

(x-\-2){x‘  — 2X+4) 

{x-\-a)  (x^+ax+4a‘) 

(3’+4)(/— 43'+i6) 

(r+2)(r+4)(r-5) 

im+2y(m+^) 

{x+i){x-s)ix+4)(x-5) 

ix+i)ix-2)ix-s)(x+4). 


§ 265 


I,  2,  3 

7* 

3,  i+l/-3,  i-l/' 

-1,3, 

“4 

8. 

1 

1 

M 

1 

1 

-2,  2, 

“4 

9- 

2,  2,  3,  4 _ 

-3,  3, 

-5 

10. 

I,K-I=^T/2i) 

-4,  5, 
2,  -2, 

+6 

— 2 

II. 

No  real  roots. 

Miscellaneous  Exercises 
§268 

1.  (m+n){m—n) 

2.  {2a+sb){2a—sb) 

3.  (i-fr")(i+/-)(i-0 

4.  y{x-\-y){x-y) 


-JI?] 


Factoring,  Quadratics,  Radicals 


147 


5 . {u+v){u^—uv+v'^) 

6.  {c^-\~d^-\-cd)  {c^-\-d^—cd)  {c-\‘d)  (c—d) 

7 . {a-\-b){a^—a^b-\- a^b^ — ab^ + b^) 

8.  2(i5+a)(is-a) 

9- 

1 0.  ( — a^b^ -\-b^){a^—ab+b''){a+b) 

11.  {x^+y^) {x^—x^y^-\-y^) 

12.  {m^+n^) {m-\-n) {m—n) {m^+n^^mn) 
{nf+n^-\-mn) 

13.  {w^-{-2W+2){w^-'2W-{‘2) 

14.  a^{ab-2>y 
15*  {x^+z^Yy^ 

16.  l^mn^{l—x—7ri) 

17.  {y^(^a){y-^a) 

18.  5(^+3)(*-“i) 

19.  3((z~io6)(a+76) 

20.  (3W-“4;z)^ 

21.  (2x+33;)(2.T+i33;) 

22.  y{b-\-(iy){c-{-y‘^) 

23.  (w— 36— 2) (w— 36+2) 

24.  (9^-s)(8^+9) 

25*  (x+2)(x^  — X— l) 

26.  (3a“+56*')(2a''~36*‘) 

27.  c:) 

28.  5(2a"~60(a"+S^') 

29.  (a^+256^)(a^+^>0 

30.  (6^:~S(/)(sc+4^/) 

3 1 . (3 +a6V)  (9 — 3a6'*c^+a^64^^) 

32.  (a+4)  (a^  4^^ + 1 — 64^+ 256) 

33.  r(2x+7)(5x-i) 

34.  {a-b-c){a-b+c) 

35.  (a+6)(a-6+i) 

36.  (io--x)(ii+x) 

37.  (2X"-7xy-3>;2)(2:^2+7A;)r-33;») 


148 


First-Year  Mathematics  Manual  [pp.  317- 


38.  (c-x)ia-7b) 

39. 

40.  (x-2)(:»:+2)(rK-3)(x+3) 

41.  (8a;— 9y)(3:c+8y) 

42.  [3(a+6)+4a;][2(a+i)— 3a;] 

43.  {sac-\-d)iisac-d) 

44.  {m-Y2,-^x-\-2y){m-\-3—x—2y) 

45-  (4y+i)(3/-2) 

46.  (x+i)(3a:;^— a:— i) 

47.  (a+i)(a^— a6+J*+i) 

48.  {a-\-b~\~d)ia~\~b — c}{a — 6“j“c)(6  d-l-c) 

49.  (3W+8M+3a— 86)  (3w+8«— 30+86) 


50- 

(a;-3.'y-sz)". 

§ 269 

I. 

y=c 

8. 

2. 

2a^  d 

.,_-3=^l/- 

9* 

t/  — 

iic^  ’ 5^ 

2 

3- 

t=i2a,  2a 

10. 

7£;=  =‘=2]//'^ 

w = 2±i/s 

II. 

4- 

12. 

See  Errata, 

5- 

p.  ISO. 

13* 

m = 2 

6. 

« = 3>  “4,  5 

n = 3 

7- 

r+s 

14. 

i(a  . b . i 
x—-! — — [-■ 
aw  w ^ 

^ a — 6 

6‘')=fc  i/'^w^(o''— 6'*)^— 4a6w(o^— 6^) 

2an 

(a—b)(2b—a) 
b^-\-ab — 0+6 


17- 

18. 

19. 


w=-f| 


x = o,  4 

ac  = i=t|/-^  or 


i=^2l/-5 


20.  a(:=i,  =*=]/2. 


3 


Factorings  Quadratics  ^ Radicals 


149 


-320] 


2. 


x+i 

x—i 

c—d 

7-Vd 

2m-s 

7W-3 


§ 270 


(a—b—c) 
— c) 

x-Ay 

x+^y‘ 


3X 


{x-z){x-2) 


§271 


rm 


2. 

2^+36 

1 

6 

M 

6(a— 6) 

I— 

II. 

a—x 

3* 

(w+4)(x-5) 

_ y(y4-3a) 

a(3;+2a) 

4- 

3;— 6 

y-4 

I 

r 

If 

4(>'+2) 

m 

5* 

f*— 16 

6. 

g=“+2 

(r— i)(r— 2)(2r- 

7- 

{oc‘-y‘y 

15.  a^-\-a^-\-a^-\-a+\ 

8. 

I 

I.  — 


2. 


g+i 

o"(a+3) 

2(a:-3) 

*-5 


I.  — days 
r+j 


§273 
^ ii-S 


5y-6 


5.  m 

6.  - 


(y+3)(3y+2)(2y+i) 

§274 

. fgh 


gh-\-hf-\-fg 


days 


First  Year  Mathematics  Manual  [pp,  320- 


ISO 


3.  6 = width 

B has  5i+^m+/+m) 
I+m 

7 = length 

8 = height 

6. 

16  barrels. 

4.  a(a^— a6+6'') 

k 

6(a^— a6+6^) 

7- 

minutes 

n — m 

\ ha-  ^(Sk+km+l+m) 
i+m 

8. 

n=2>. 

ERRATA 


P.  288,  No.  4,  second  paragraph,  third  line,  for  *^of’^  read 


P.  297,  § 232,  No.  9,  should  be 


64a^—g6a'^b+36ab‘^  ^ 
4a^— i8a6 


i6a^— I2a6 
8a^ — 7 2 + 1 6 2 

P.  302,  § 238,  No.  14,  should  be  24—103^—24. 

No.  24,  ''  k^^-2ok^L^^"-6gL^^\ 

P.  317,  No.  28  should  be  10^4+45^^6^  — 2564. 

P.  318,  No.  12,  problem  is  impossible  since  the  condition 
cc 

— ^=0  renders  2 of  the  denominators  o. 

3 


P.  319,  § 272,  denominator  should  be 


5 


CHAPTER  XIV 

POLYGONS,  CONGRUENT  TRIANGLES,  RADICALS 

General  suggestion, — To  expedite  the  blackboard  work 
of  the  class,  the  teacher  may  write  out  the  problems  and  the 
theorems  to  be  done,  on  slips  of  paper,  or  cards,  before  the 
recitation  period,  and  hand  slips  to  certain  pupils,  or  have 
pupils  draw  one  each  from  the  teacher’s  hand.  Pupils  will 
then  go  at  once  to  the  board  and  solve  the  problem  or  prove 
the  theorem. 

In  this  chapter  at  least  four  results  should  be  aimed  at: 

{a)  The  acquirement  of  a considerable  number  of  geomet- 
rical concepts  and  laws. 

{b)  Skill  in  deriving  and  proving  new  laws  from  those 
already  known. 

(^;)  Greater  freedom  in  the  use  of  the  algebraic  equation. 

{d)  Further  use  of  radicals  in  approximations  of  the 
value  of  geometrical  lines. 

Lesson  i : to  Problem  i6,  page  326 

§ 275.  Develop  Problem  i orally  with  the  class. 

As  the  sum  of  the  angles  of  each  triangle  is  180°,  the 
sum  of  the  angles  of  a polygon  composed  of  5 triangles  is  5X 
180°.  The  sum  of  the  angles  of  the  polygon  is  the  same  in 
value  as  the  sum  of  the  angles  of  the  5 triangles  combined 
except  for  the  value  of  4 RZ^  about  P.  Hence  the  sum  of 
the  angles  of  the  polygon  is  5X 180°,  less  360°,  or  540°. 

Have  the  class  learn  the  definitions  of  § 276  and  § 277. 

Let  teacher  work  out  with  class  some  of  Problems  i to 

§ 278,  assign  others  as  home  work — some  may  be  given 
out  during  succeeding  lessons  as  home  work. 

Problem  12.  6:^:  = (6  — 2)180  = 720,  x=i2o,  etc. 

151 


152 


First-Year  Mathematics  Manual  [pp,  J26- 

Lesson  2:  to  ^ 254,  page 

Problem  16.  Give  the  class  opportunity  to  state  how 
it  can  be  determined  whether  tiles  of  certain  shapes  will  lay 
a floor. 

The  angles  of  the  polygon  must  be  contained  exactly 
in  the  space  of  4 RZ^.  The  angle  of  a 3-side  is  60°,  so  6 
tiles  will  cover  the  space.  Four  4-sides  will  cover  it.  The 
angle  of  a s-side  is  108®  and  is  not  contained  exactly  in 
360.  The  angle  of  a 6-side  is  120°,  and  3 will  fill  the  space. 
The  8-side  (135°)  and  the  is-side  (156°)  cannot  be  so  used. 

Problem  17  depends  on  the  law  that  the  sum  of  the  three 
angles  of  a triangle  is  180°.  Go  orally  over  Problems  17  to 
21.  See  that  the  pupils  understand  and  learn  the  theorem 
following  Problem  21. 

Develop  orally  § 279,  Problem  i. 

For  Problem  2 see  Fig.  62,  page  56.  Solve  Problems  2 
and  3 with  the  class.  Assign  Problems  4 to  9 for  home  work, 
also  definitions  of  §§  281,  282,  and  283. 

Lesson  3 : § 284  to  § 290 

Solve  orally  §§  284  to  289  Problem  i. 

Have  the  class  draw  triangle  of  Problem  2 on  paper  in 
class.  Have  them  cut  the  triangle  out  or  use  tracing  paper. 

Teacher  work  Problem  3 on  board,  and  go  carefully  over 
Problem  4 with  the  class.  Hold  them  prepared  to  do  it 
next  day  and  also  to  state  it  as  a theorem,  as  in  § 290. 

Lesson  4:  through  Problem  10,  page 

Solve  orally  with  the  class  Problem  i under  § 290. 

6+6'  is  a straight  line,  for  the  sum  of  the  two  adjacent 
angles  is  2 RZ^  or  a straight  angle.  The  entire  figure  is  an 
isosceles  triangle  because  c = c by  hypothesis.  Assume 
that  the  angles  opposite  the  equal  sides  of  an  isosceles 


-340]  Polygons y Congruent  Triangles , Radicals  153 

triangle  are  equal  (see  also  top  of  page  344).  Then  the 
remaining  angles  are  equal,  triangle  acb'  ^ triangle  acb 
having  the  side  a and  the  two  adjacent  angles  of  one  equal 
to  the  side  a and  the  two  adjacent  angles  of  the  other. 

Problems  3 to  8 are  proved  by  theorem  § 290. 

Assign  Problems  9 and  10  for  home  work. 

Lesson  5 /t?  § 293,  page  337 

Teacher  go  through  Problems  ii  and  12  with  the  class 
and  have  them  prepared  to  do  Problems  ii,  12,  and  13  next 
day,  and  be  able  to  state  the  theorem  of  § 291,  also  § 292, 
I,  II,  and  III. 

Lesson  6:  through  Problem  4,  page  338 

Let  the  teacher  draw  Fig.  267  on  board.  State  the 
theorem  and  see  if  class  can  prove  it.  It  is  clear  that  0 P, 
OB,  and  Z Oin  AP O B equal P D, O A,  and  ZOin  AP 0 A. 
Assign  Problem  2 as  home  work,  either  to  be  handed  in  writ- 
ten out,  or  to  be  recited  orally. 

Teacher  will  go  over  Problem  3 if  there  is  time  and  assign 
Problems  3 and  4 for  home  work. 

Lesson  7 : from  Problem  5,  page  338,  to  middle  of  page  341 
{Polygons) 

Teacher  to  develop  Problem  5 with  class.  Assign 
Problems  6 and  7 for  home  work.  Talk  over  § 294  with  the 
class. 

Get  class  to  prove  Problem  i.  Note  that  the  perpendic- 
ulars are  the  lines  on  which  the  distances  of  P from  the 
sides  of  the  angle  are  measured. 

To  prove  the  triangles  congruent,  use  theorem  after 
Problem  21,  page  327,  as  ZD=  ZC  a,sx  = y,  .*.  z = w.  Then 
use  § 292,  II. 


IS4 


First-  Y ear  M athematics  M anual  [ pp . 340- 


Problem  2,  page  340,  may  be  assigned  for  home  work  as 
practice  in  construction  with  compasses.  Teacher  will  go 
over  Problems  3 and  4 with  class,  to  be  recited  next  day. 
In  Problem  4 use  theorem  at  top  of  page  340,  and  the 
equality  axiom. 

Lesson  8:  from  Polygons,  page  341,  to  Problem  6,  page  343 

Talk  over  definition  of  polygons  with  class  and  compare 
§ 275,  page  323.  Develop  Problems  5,  7,  and  8 with  class. 
§ 295,  page  342,  Problem  i,  have  class  make  the  construction 
of  perpendicular  from  a point  C on  a line,  and  give  a proof 
by  congruency  of  triangles.  Use  § 292,  III.  F C is  perpen- 
dicular because  the  straight  angle  E C D is  divided  into  two 
equal  parts  by  F C. 

In  same  manner  bring  out  the  points  of  proof  for  Problems 
2,  3,  4,  5,  dependent  on  § 292,  page  337.  Class  to  be  pre- 
pared to  give  these  proofs  next  lesson. 

Lesson  9:  Problem  6,  page  343,  through  page  344 

Problems  6,  7,  8,  and  9 to  be  developed  in  class.  Prob- 
lem 7 leads  the  theorem  at  top  of  page  344,  which  is  to  be 
learned  for  future  use. 

In  Problem  9 use  theorem  after  Problem  21,  page  327. 

Assign  Problem  10  to  part  of  class,  and  Problem  ii  to 
the  rest,  for  home  work.  Require  all  to  know  proof  for 
Problems  6,  7,  8,  and  9 at  next  lesson. 

Lesson  10:  pages  345-48 

By  oral  work  on  Problem  12  and  § 296,  Problem  i,  bring 
out  clearly  hypothesis  and  conclusion.  Have  pupils  learn 
them  through  using  them  in  written  demonstrations. 

Assign  Problems  2 to  21,  3 or  4 each  to  different  pupils 
for  written  home  work,  and  have  them  put  on  board  one 
by  each  pupil  next  day. 


-349]  Polygons y Congruent  Triangles ^ Radicals 


15s 


In  Problem  3,  page  346,  the  figure  formed  is  a parallelo- 
gram, the  opposite  sides  are  equal  (Problem  2,  page  34S)- 

In  Problem  4,  reference  should  be  to  Problem  8,  page 
334,  not  12  as  stated  in  the  text. 

Problem  5 is  proved  from  Problem  2,  page  345. 

Problem  6,  D O = 0 B from  Problem  4 above.  Z D 0 C = 
Z B O C because  the  triangles  of  same  lettering  are  con- 
gruent, 3 sides  of  one  = 3 sides  of  the  other,  each  to  eacho 

Problem  7,  use  § 292, 1. 

Problem  8,  Problem  12,  page  345,  could  be  used  instead 
of  superimposing. 

Problems  9 and  10.  These  follow  from  Problem  8. 

Problem  11,  c = a,  opposite  angles,  then  a = e. 

Problem  13.  This  follows  from  Problem  ii  preceding, 
and  § 106  {b),  page  138. 

Problem  15  follows  from  Problem  13. 

Problem  16  follows  from  Problem  14  and  § 106  (J),  page 
138. 

Problem  17  follows  from  § 106  {b)  and  Problems  15  and 
16. 

Problem  18,  ABC=ACD,  3 sides  mutually  equal. 
Then  the  alternate  interior  angles  are  equal  and  the  lines 
parallel. 


Lesson  ii:  page  34g  to  §301,  page  351 

§ 297.  Problems  i to  5 can  be  assigned  tor  home  work, 
I or  2 to  each  pupil.  Teacher  mav  develop  Problem  4 with 
class,  hi.  BO  from  Problem  2 preceding,  hence,  10^  = 5^+ 
A",  /j"  = io"-s",  /j"  = 7S,  ^ = 51/5-  Area  = 5 • 51/5  = 251/5. 
Problem  5.  (i)  Altitude  = 1/^,  area  = 31/^27  = 91/3 

(2)  /g=i/48,  area =41/48 

(3)  h — V 108,  area  = 6P  108 

(4)  A = i/^,  area=|T/5 

(5)  A = ^rea=|l/-^;j^. 


First-Year  Mathematics  Manual  [pp.  J4g- 


156 


Teacher  to  go  over  §§  298,  299,  and  300  with  the  class. 
Assign  four  or  five  parts  of  Problem  2,  § 300,  to  each  pupil 
for  home  work. 


Lesson  12 : § 301,  page  351,  to  § 303,  page  353 

After  working  on  board  one  or  two  parts  of  Problem  i, 
§ 302,  assign  five  or  six  selected  parts  of  Problems  i and  all 
of  Problem  2 to  each  pupil  for  home  work. 

After  the  results  of  Problem  2 are  verified  at  next  lesson 
have  pupils  copy  on  fly-leaf  of  textbook  these  approximate 
square  roots  for  future  use,  and  later  add  to  the  table  the 
square  roots  of  10,  ii,  13,  14,  15.  It  may  be  well  also  to 
have  pupils  make  a table  of  squares  from  i to  30.  Both 
tables  can  be  copied  into  the  second-year  text  later. 

Develop  with  class  § 302,  Problem  i,  assign  three  or 
four  parts  of  Problem  2,  page  352,  to  each  pupil  for  home 
work.  Caution  the  class  that  the  polynomials  should  be 
arranged  and  remainders  arranged,  according  Lo  the  powers 
01  some  letter. 


Lesson  13:  § 303  to  Problem  2,  page  356 

Develop  with  class  § 303,  also  § 304,  Problems  i,  2,  3,  4. 

Assign  Problem  5 for  home  work  and  verify  answers  next 
day. 

Develop  § 305,  Problems  i,  2,  3,  and  emphasize  the  differ- 
ence between  the  square  root  of  a sum  and  that  of  a product. 

Give  Problems  4,  5,  6 for  home  work,  also  § 306,  Prob- 
lem I. 

Problem  4,  page  355,  first  part,  3^  = s^  — {sl2y  = ls^  s^  = 
V or  12.  5 = 3.52+,  area  = fX3  S2  = S-28. 

Problem  5.  5^  = 5'*--(5/2)^  = 35V4.  5^=  100/3,  etc. 


Polygons^  Congruent  Triangles,  Radicals  157 

Lesson  14:  Problem  2,  page  356,  to  Problem  21,  page  jjS 


Teacher  ask  class  how  they  would  do  Problem  2,  then 
show  the  two  methods  ot  § 307.  The  first  is  less  exact 
because  the  divisor  is  only  approximately  correct,  as  also  the 
dividend.  Do  Problem  2,  page  357  with  the  class.  Get 
pupils  to  stale  the  method,  then  require  them  to  memorize 
§ 308.  Assign  to  each  pupil  four  or  five  of  the  fifteen 
examples  in  § 308,  page  357,  for  home  work.  Verify  results 
of  all  fifteen  at  next  recitation.  Use  Problems  16  and  17  and 
18  for  test  on  previous  work.  In  Problem  19  have  one  pupil 
find  h on  the  board,  and  another  find  s in  terms  of  h.  Show 
how  (3)  and  (4)  may  be  used  in  solving  Problem  20  and 
assign  parts  for  home  work. 

Lesson  15:  Problem  21,  page  358,  and  Problem  p,  page  360 

In  Problem  21  is  shown  how  best  to  approximate  values 
when  a radical  is  in  both  numerator  and  denominator  of  a 
fraction.  Assign  a few  parts  of  Problem  22  to  each  pupil 
for  home  work;  also  4 or  5 of  the  problems  in  the  rest  of 
the  lesson  to  each  pupil.  Suggest  the  saving  in  time  by  use 
of  formulas  already  proved,  such  as  h,  s,  and  c on  pages 
358  and  359,  whenever  they  apply.  There  is  an  error  in  the 
formula  in  Problem  8,  page  360.  It  should  be  changed  to 


Have  pupils  insert  the  exponent  of  R, 


Lesson  16:  page  361  to  Algebraic  Problems,  page  362 

Assign  5 or  6 of  these  problems  to  each  pupil  for  home 
work. 

Suggested  groupings,  10,  ii,  12,  13;  14,  15,  16;  18,  19, 
25;  20,  21,  22;  20,  21,  23;  20,  21,  24;  call  attention  to  the 


158  First-Year  Mathematics  Manual  [pp.  362- 


fact  that  in  Problem  10  the  radius  is  the  altitude  of  the  tri- 
angle. Use  formula  A • 

In  the  equation  — both  members  may  be 

divided  by  1/3,  and  the  radical  removed. 

Problem  ii.  (page  358,  19  [4]) 

4 4 


The  hexagon  is  six  times  the  triangle,  hence  the  hexagon 


Problem  12.  A=-^^  ; multiply  by  6, 


Problem  14.  s^  = 2R^,  s = Ry^2,  A = {RV2y  = 2R^, 


Problem  15.  ^=5*  (the  square  of  a side)  as  s = R\/2, 
then  R — sJ\/2  or 

Problem  17.  Last  part.  s = V^A,  R = V' AI1/2  (see 
Problem  13)  = l/I/l/2Xl/2/l/2==  1/^/2  or 

Problem  i8.  Let  x=s, =R,  but  R=sl\/2  or  xj\/2-, 

2 

, , 

.*/ = rr/|/2,  multiply  by  21/2,  x\/ 2 + 21/ 2 = 2:r.  Divide 

2 

— 2 

by  1/2,  x+2  = \/2X,  2 = \/2X—X.  82-f-. 

Then  or  ^ = 23 . 23+ 


-J^i]  Polygons j Congruent  Triangles,  Radicals 


159 


Problem  19. 


Substituting  b for  2 in  Problem  18, 


x+b 

2 


jr/ 1/2,  then  x=- 
Problem  20. 


b 

1/2  — 1 ’ 

A = ^b  • h = xy. 


Problem  21.  The  two  triangles  = 2:^3^,  from  Problem  20. 

Problem  22.  a;^  = i5^  — 12^  = 81.  ,\x  = g, 

^ = 2 • 9 • 12  = 216. 

Problem  23.  x^  = i44  — 81  = 63.  ^ = 3l/7* 

A=2  • 9 • 31/7  = 541/7  = 142.867. 


Problem  24.  Let  x be  smaller  diagonal,  then  2X  is  the 
other.  A=x^,  half  the  product  of  the  diagonals  (see  Prob- 
lem 21,  page  361).  Then  128=^1;%  and  ^ = 81/2,  2x  = i6\/2, 
The  diagonals  are  11.31  and  22.627. 

Problem  25.  (i)  225=x^+x^=2:r%  x=i^\^\. 

/l  = i5T/iX-#l^i=^-^  = 56j.  Note  the  result  (^f^)  is 
\ the  square  of  the  hypotenuse. 

(2)  - ^ - = -®:^=  15! . Consider  the  hypotenuse  to  be  the 


diagonal  of  a square.  Then  use  Problem  21,  page  361,  and 
we  have  ^(31/7)*  for  the  area  of  the  rhombus  (square),  ^ of 
that  for  the  triangle  (^  of  ^ is  J). 

(3)  Similarly  h^j 4. 


Lesson  17:  page  362  to  end  of  the  Book 

These  problems  may  be  made  to  compose  a lesson  or 
two,  or  may  be  given  as  tests  or  review,  a few  at  a time. 
They  comprise  simple  equations  in  one,  two,  or  three  un- 
knowns, and  complete  quadratic  equations.  Page  365 
gives  problems  in  ratio  and  proportion  to  form  the  equations 
in  three  unknowns. 


i6o 


First-Year  Mathematics  Manual  [pp,  32^- 


Note  the  following  errors  in  the  first  edition : Problem  2, 
(3)  b' c'  = 16(2+6);  Problem  4,  (2)  a=2r-\-2\s 

2S 

(or  2 . 5s).  (3)  a'  = 5r+“  — i;  Problem  8.  ii+y  (instead 

ot  12— y)  and  ^in  place  of  i — _ 

§ 309.  As  the  ratio  of  similarity  is  3 then  a:+2=3(io+ 
y/3)=30+y,  and  3*— 2=12+2;,  72+22;  = 140+y,  and  so 
for  all  five  problems. 


Chapter  XIV,  Answers 

Page  325: 

3-  90 

4.  360,  540,  720,  180 

5.  900,  1,080,  1,440,  2,340,  2,880 

6.  (i)  180,  1,260,  2,520,  (s—2)  180;  (2)  7,  42 
8.  22,97,3 

10.  42,  93,  3 

Page  326: 

11.  108 

i^n 

12.  120, 135, 158H.  — 

13.  128+ 144, 156 

14.  6 

IS-  8,  9,  10,  36 
16.  3,  4,  6. 

Page  328: 

3-  16  7-  45 

4.  20  8.  360/y 

5.  2(a+i)  9.  18. 

6.  24 


— 2)180 
n 


~354\  Polygons,  Congruent  Triangles,  Radicals 


i6i 


Page  334: 

9.  x = 4 or  I 
19  or  4 

Page  335: 

B C,  A B,  i9ff  . 

Page  349: 

4-  A = 1/75,  ^=51/75  _ _ _ _ 

5.  1/27^1/ 48,  V 108,  V 31/ 27,  41/ 48, 

Page  350: 

§ 299.  I.  2,  2,  2;  I,  3,  3,  3. 

Page  351: 

2.  (i)  28,  (2)  39,  (3)  47,  (4)  49,  (5)  206,  (6)  229,  (7)  315, 

(8)  336,  (9)  347.  (10)  531.  (ii)  624,  (12)  718. 

§ 301. 

1.  (i)  4.4,  (2)  1.7,  (3)  2.3,  (4)  10.7,  (5)  14.5,  (6)  1.26, 
(7)  1-46,  (8)  3. II.  (9)  3.33,  (10)  .35,  (ii)  .19,  (12)  7.16, 
(13)  -73.  (14)  8.86,  (is)  .992 

Page  352: 

2.  1 .4142,  1 . 7320.  2 . 2360,  2 . 4496,  2 . 6457. 

§ 302. 

2.  (i)  Sr-^s,  (2)  x‘-3x->r2,  (3)  X+2J-+3Z,  (4)  3a- 
b^-c,  (s)  x‘-\-x+i,  (6)  i—a^+a,  (7)  3Z-2a+c,  (8)  r-r^-3, 

(9)  s^-r^-j-i,  (10)  2x^—y-{-z^ 

Page  354: 

5-  (i)  3t/5._(2)  51/2.  (3)  21/7,  (4)  4l/^(s)  54/5.  (6) 
101/2,  (7)  121/2,  (8)  8xi/2a;y,  (9)  isx*3'5i/3j;,  (10)  (fl— &) 
4/7,  (ii)  xia—byv'^ax—^bx,  (12)  3V ^a^—zl^,  (13)  2;(s^— 
2<")4/3.  (14)  a/^l/S- 


i62  First-Year  Mathematics  Manual  {pp,  J55- 

355: 

4.  3.464,  10.39,  13-856,  17-32,  20.785 
5.20,,  46.76,  83.136,  129.9,  187.065 
7*  it  it  it  'h't  T^>  tV;  f ; it;  Ht  if;  it* 

Page  35(>' 

2 a ax  4 z{a—b)  a—x  m-\-n 

a’  be’  by’  m—n’  $(a+b)’  c+y’  io{a+b)’ 
2x—y  i2{c+2d)  4ab^c(m—ny  (6a;— 5y)^(a+3<^)^ 
Sy+z’  4(2x-sy)’  gc^d(r+sy  ’ (sa-cy(2r-5s) 

2.  -5773,  -8165,  .7745,  -8451- 

Page  357- 

§ 308. 


I-  -7745 

9-  *3779 

2.  .8660 

10.  2.0413 

3 2.6457 

a 

11.  1.3093 

12.  1.2649 

2.4496  \b 
4-  b \ 

■3- 

5*  -7071 

6.  .5773 

7.  1.7880 

8.  .9354 

■5- 

Page  358- 

16.  8.0826 

2-3093 

9-2373  11.5460 

25.4026 

28.2891 

2 . 3093 

36-9492  57-7300 

279.4286 

18.  6.928 

10.392 

12.456  11.546 

16.165 

20.784 

46.764 

76.136  57-733 

113-157 

20.  5.196 

8.66 

7-794 

6.062 

15-588 

43-3 

35-073 

21.217 

-jdi]  Polygons,  Congruent  Triangles,  Radicals 


163 


Page  359: 

22.  (i)  1 . 1368 

(2)  .2886 

(3)  -7215 

(4)  -6236 
23-  3-651 

24.  2.291. 

2*  7-6,  8, 

S. 


(5)  • 1924 

(6)  7-3421 

(7)  . 24496a*6 


/ON  * 

(8)  I -4142- 

(9)  2a 


Page  360: 

4.  7.141,  7.78,  VAi/s 
5-  15-19,13-16 
6.  120° 


7-  3 

8- 


9.  10.392,  2.598,  23.382,  40.868,  584.55 


3f^3 


Page  361:  

10.  1.4142,2.95.3.02,^1/2/41/3 

2T  I 

11.  A = 2r^j/3, 

o 


15- 

17- 

18. 


A=s\  R=-^ 

1/2  _ 

17.68  10,  lyi/^,  or  .jojy,  \V 2k 

23-23 

h b 


19. 


.4142 


164 


First-Year  Mathematics  Manual 


Page  362: 

22.  18,  216 

23.  142.867 

24.  8|/2  or  11.3136,  161/2  or  22.627 

¥ 

25-  S6i  iSi--  I-  c = 8,  a;=io,  3;=i5. 

4 

Page  363: 

2.  (i)  2:,  12;  y,  ii^;  z,  20;  sof;  c,  no;  A,  48. 

(2)  X,  4;  y,  7f ; z,  -i;  b,  iif;  A,  -3;  c,  95. 

(3)  2:,  5,  7;  y,  7,  -3;  z,  -24,  4;  6,  42,  12;  c,  480; 

IS,  35- 

Errors,  b' =3{y-]r^)‘,  c'  = i6(z+6). 

3.  0 = 12,  c=3,  2;=9 

4.  (i)  5=5,  >'=16;  (2)  5 = 12,  r=i5.  Error  in  (2): 

2S 

0=2^+255;  Error  in  (3):  a'  = sr+y— i;  s=  — if;  r=\ 

Page  364: 

5-  5,  I,  3 

6.  “I4tV5,  ~3tS^ 

7-  -77,  -385,  1078 

8.  Errors: 

12— y should  be  ii+y. 

should  be 

4 4 

:x:  = 2,  — I 

Sides  10,  I. 

Page  365: 

1.  14,  o,  16  4.  3,  4,  2 

2.  3,  3,  2 5-  5,  2,  I. 

3-  3,3,  2 


UNIVERSITY  OF  ILUNOI9-URBANA 


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